Piecewise Graphs and Discontinuities | Malone Math

The Central Idea

A piecewise-defined function or relation uses different rules on different parts of its domain. Its graph is assembled from those pieces.

The central task is not merely to graph each formula. We must also determine where each formula applies and what happens where the pieces meet.

Those meeting points may be smooth, may form corners, or may contain holes, jumps, or other discontinuities.

One Object, Several Rules

A single function can require more than one algebraic rule. For example,

\[ f(x)= \begin{cases} x+2, & x<1,\\ x^2, & x\ge 1. \end{cases} \]

The first rule is used only when \(x<1\). The second rule is used only when \(x\ge 1\).

Reading Piecewise Notation

In a piecewise definition, every row contains two parts:

a formula or output rule;
a condition telling us when that rule applies.

The condition is part of the definition. Ignoring it changes the function.

The Conditions Partition the Inputs

Well-designed piecewise conditions divide the intended domain into regions. For a function, each allowed input should receive exactly one output.

Conditions may use intervals such as \(x<0\), \(0\le x<3\), and \(x\ge3\), or they may describe discrete sets such as \(x\in\mathbb Z\).

A Piecewise Definition Can Still Describe a Function

Having several formulas does not violate the definition of a function. What matters is that each input in the domain corresponds to only one output.

The vertical-line test therefore applies to the completed graph exactly as it does to any other graph.

Overlapping Conditions Require Care

Suppose two rows both apply at \(x=2\). If they produce different outputs, the notation does not define a single-valued function at that input.

Overlap is harmless only when the overlapping rules agree wherever both are active.

A Gap in the Conditions Removes Inputs

If no row applies to an input, that input is not in the domain unless another part of the definition supplies a value.

For example, conditions \(x<1\) and \(x>1\) leave \(x=1\) uncovered.

Graph Only the Allowed Part of Each Rule

If the rule is \(y=x+2\) for \(x<1\), do not draw the entire line. Draw only the portion whose \(x\)-coordinates satisfy \(x<1\).

A piecewise graph is a collection of restricted graphs.

Endpoints Carry Information

At a boundary between pieces, the graph must show whether the boundary input belongs to a piece.

A closed point indicates inclusion.
An open point indicates exclusion from that particular piece.

Strict Inequalities Produce Open Endpoints

A condition such as

\[ x<3 \]

excludes \(x=3\). The corresponding endpoint is drawn open when the graph reaches that boundary.

Inclusive Inequalities Produce Closed Endpoints

A condition such as

\[ x\le3 \]

includes \(x=3\). The corresponding endpoint is drawn closed.

Example: Two Linear Pieces

Consider

\[ f(x)= \begin{cases} x+1, & x<2,\\ 5-x, & x\ge2. \end{cases} \]

For the first piece, the boundary output would be \(3\), but \(x=2\) is excluded, so \((2,3)\) is open for that piece.

For the second piece,

\[ 5-2=3, \]

and \(x=2\) is included, so \((2,3)\) is closed.

Different Rules Can Meet at the Same Point

In the previous example, the two formulas are different, but both approach the same point \((2,3)\).

A change in formula therefore does not automatically create a break in the graph.

A Formula Boundary Is Not Necessarily a Graphical Break

Piecewise notation is an algebraic organization. Continuity is a graphical and analytical property.

The formulas may change at a boundary while the graph remains connected and continuous there.

Example: A Continuous Piecewise Function

Let

\[ g(x)= \begin{cases} 2x+1, & x<1,\\ x+2, & x\ge1. \end{cases} \]

At \(x=1\), both rules give the boundary value \(3\):

\[ 2(1)+1=3, \qquad 1+2=3. \]

The pieces meet at \((1,3)\), so the change of rule does not create a discontinuity.

Example: A Jump Between Pieces

Consider

\[ h(x)= \begin{cases} x+1, & x<2,\\ x+4, & x\ge2. \end{cases} \]

The left piece approaches \(3\) at \(x=2\), while the right piece begins at \(6\). The graph jumps from one height to another.

What Continuity Looks Like Graphically

Informally, a graph is continuous over an interval when it has no breaks there and nearby inputs produce nearby outputs.

The familiar phrase “draw it without lifting your pencil” is useful intuition, but precise continuity also depends on the domain and on what happens at individual points.

Continuity at a Point

At an interior point \(x=c\), continuity requires three ideas to agree:

the function has a value \(f(c)\);
the graph approaches a single height as \(x\) approaches \(c\);
that approached height equals \(f(c)\).

Later analysis gives these ideas a formal limit notation. Graphically, we can already inspect them directly.

Left-Hand and Right-Hand Behavior

Near a boundary \(x=c\), inspect the graph from both sides.

The left-hand behavior describes points with \(x
The right-hand behavior describes points with \(x>c\).

If the two sides approach different heights, the graph has a jump at that boundary.

The Function Value Can Differ from Nearby Behavior

A filled point records the actual value of a function at an input. Nearby points describe how the graph behaves around that input.

Those two pieces of information need not agree.

Example: Defined but Discontinuous

Suppose the graph follows \(y=x+1\) near \(x=2\), except that the point \((2,3)\) is open and a filled point is placed at \((2,7)\).

Then

\[ f(2)=7, \]

even though nearby graph values approach \(3\). The function is defined at \(x=2\), but it is not continuous there.

A Discontinuity Does Not Always Mean Undefined

A discontinuity is a failure of continuity, not merely a missing function value.

A function can have a perfectly defined value at a point and still be discontinuous there because the surrounding graph does not meet that value.

Removable Discontinuity

A removable discontinuity occurs when the surrounding graph approaches one finite point but the function either omits that point or assigns a different value there.

Graphically, it commonly appears as a hole.

A Hole

An open point inside an otherwise continuous-looking curve indicates that the corresponding point is excluded.

If no other filled point occurs at the same input, that input is absent from the domain.

A Hole Can Be Filled Somewhere Else Vertically

At the same \(x\)-coordinate as a hole, a function may contain a filled point at another height.

The input then remains in the domain, but the graph is still discontinuous because its assigned value does not match the nearby behavior.

Example: A Removable Discontinuity from Algebra

Consider

\[ r(x)=\frac{x^2-4}{x-2}. \]

For \(x\ne2\), factoring gives

\[ r(x)=\frac{(x-2)(x+2)}{x-2}=x+2. \]

But the original denominator is zero at \(x=2\), so \((2,4)\) is a hole, not a filled point.

Simplifying Does Not Restore an Excluded Input

The simplified formula \(x+2\) describes the same outputs only where the original expression was defined.

Therefore,

\[ \frac{x^2-4}{x-2}=x+2 \quad\text{for }x\ne2, \]

not for every real number.

Jump Discontinuity

A jump discontinuity occurs when the graph approaches two different finite heights from the left and right.

No single point inserted at the boundary can make the two sides meet.

Example: Step-Like Jump

Let

\[ s(x)= \begin{cases} 1, & x<0,\\ 4, & x\ge0. \end{cases} \]

The graph approaches \(1\) from the left and \(4\) from the right. The jump size is

\[ 4-1=3. \]

Jump Size

When both one-sided heights are finite, the vertical separation between them can be measured.

If the left side approaches \(L\) and the right side approaches \(R\), the magnitude of the jump is

\[ \boxed{|R-L|}. \]

Infinite Discontinuity

An infinite discontinuity occurs when graph values grow without bound in magnitude near an input.

This behavior is commonly associated with a vertical asymptote. The next lesson develops asymptotes systematically.

Example: Reciprocal Behavior Near Zero

The function

\[ f(x)=\frac{1}{x} \]

is undefined at \(x=0\). As inputs approach \(0\), the output magnitude grows without bound. The line \(x=0\) is a vertical asymptote.

Not Every Gap Is a Single-Point Discontinuity

A graph may be absent over an entire interval because those inputs are not in the domain.

It is useful to describe the visible graph as having a gap, but formal continuity statements must respect the actual domain rather than treating every excluded input as an ordinary interior point of the function.

Endpoints and One-Sided Continuity

At an endpoint of the domain, there may be graph points on only one side. Continuity there is judged using the side that belongs to the domain.

Thus a closed endpoint can be continuous from within its domain even though no graph exists beyond the endpoint.

An Open Endpoint Is Not Automatically a Discontinuity of a Function at That Point

If an endpoint is open, the endpoint input may not belong to the domain at all. Formal continuity “at that input” is then not evaluated in the usual way because the function has no value there.

Graphically, however, the open point still records an excluded boundary and may signal a break between pieces.

Piecewise Functions Can Contain Corners

A graph can be continuous even when it changes direction abruptly.

A corner is not a discontinuity merely because the graph is not smooth there.

Absolute Value as a Piecewise Function

The absolute-value function can be written

\[ |x|= \begin{cases} -x, & x<0,\\ x, & x\ge0. \end{cases} \]

The two linear pieces meet at the origin. The graph has a corner there but is continuous.

Corners and Continuity Are Different Questions

Continuity asks whether the graph joins without a break. Smoothness asks whether its direction or slope changes smoothly.

A graph may be continuous but not smooth.

Cusps

A cusp is a sharp point where the graph remains connected but changes direction in an especially abrupt way.

Like a corner, a cusp need not be a discontinuity.

Vertical Tangent Behavior

A graph may become locally vertical while remaining continuous. Visual steepness alone therefore does not establish a discontinuity.

The question is whether the graph's values connect appropriately, not whether the slope remains finite.

Step Functions

A step function is constant on intervals and changes value at boundaries.

Its graph resembles horizontal steps, usually with carefully paired open and closed endpoints.

The Greatest-Integer Function

The floor function

\[ y=\lfloor x\rfloor \]

returns the greatest integer less than or equal to \(x\). For every integer \(n\),

\[ \lfloor x\rfloor=n \quad\text{when}\quad n\le x Each step is closed on the left and open on the right.

Step Functions Have Repeated Jumps

The floor function jumps at every integer. The discontinuities are not random; they repeat according to the piecewise structure.

The Ceiling Function

The ceiling function

\[ y=\lceil x\rceil \]

returns the least integer greater than or equal to \(x\). Its endpoint conventions are the opposite orientation from those of the floor function.

The Sign Function

A common piecewise model is the sign function:

\[ \operatorname{sgn}(x)= \begin{cases} -1, & x<0,\\ 0, & x=0,\\ 1, & x>0. \end{cases} \]

Its graph contains two horizontal rays and the isolated point \((0,0)\).

Single-Point Pieces

A piecewise definition can assign a rule to exactly one input, such as

\[ f(x)=7\quad\text{when }x=2. \]

Graphically, this contributes the single point \((2,7)\).

Discrete Pieces

Piecewise rules can also operate on discrete domains. A rule might apply only when \(x\) is an integer, or one set of integer inputs might use a different rule from another.

Do not connect discrete points unless the relation explicitly includes the values between them.

Piecewise Relations Need Not Be Functions

Piecewise descriptions are not limited to functions. A relation can combine arcs, line segments, isolated points, or other restricted pieces.

The completed relation is a function only if it passes the vertical-line test.

Graphing a Piecewise Function from a Formula

Identify every formula.
Identify the condition attached to each formula.
Graph each formula only on its permitted inputs.
Evaluate boundary coordinates.
Mark each boundary open or closed according to its condition.
Inspect how neighboring pieces meet.

Start with Boundary Inputs

Boundary inputs are often the most informative points in a piecewise graph. Evaluate the adjacent formulas there even when one side excludes the boundary.

Doing so reveals whether the pieces meet, jump, or leave a hole.

Example: Three Pieces

Consider

\[ f(x)= \begin{cases} -2, & x<-1,\\ x^2, & -1\le x<2,\\ 3x-5, & x\ge2. \end{cases} \]

The graph consists of a horizontal ray, a restricted parabola, and a linear ray. The endpoint status must be handled separately at \(x=-1\) and \(x=2\).

Analyze the Boundary at \(x=-1\)

The first piece would have height \(-2\) at \(x=-1\), but its condition is strict, so \((-1,-2)\) is open.

The middle piece includes \(x=-1\) and gives

\[ (-1)^2=1, \]

so \((-1,1)\) is closed. This boundary contains a jump.

Analyze the Boundary at \(x=2\)

The middle piece approaches

\[ 2^2=4, \]

but excludes \(x=2\), giving an open point at \((2,4)\).

The final piece includes \(x=2\) and gives

\[ 3(2)-5=1, \]

so \((2,1)\) is closed. This is another jump.

From a Graph Back to a Piecewise Formula

The process can be reversed. For each visible piece:

identify its algebraic family or equation;
determine the interval of \(x\)-values on which it appears;
translate endpoint markers into strict or inclusive inequalities;
combine the rows into piecewise notation.

Endpoint Markers Become Inequality Symbols

An open point at \(x=a\) typically corresponds to \(<\) or \(>\). A closed point typically corresponds to \(\le\) or \(\ge\).

Always read the actual domain direction before choosing the symbol.

Example: Reading Conditions from a Graph

Suppose a line segment is drawn from an open point at \(x=-2\) to a closed point at \(x=3\). Its input restriction is

\[ \boxed{-2

A Closed Point Belongs to the Whole Function

At a boundary where one piece is open and another is closed, the filled point determines the actual function value.

Do not average the two boundary heights or treat both as outputs of the function.

Open Points Describe Excluded Candidates

An open point often shows where a formula would have landed if its boundary had been included.

It is useful information about nearby behavior, but it is not itself a point on the graph.

Domain from a Piecewise Graph

The domain is the union of all input values represented by the pieces.

Open and closed endpoints, gaps, isolated points, and arrows all affect the final domain.

Range from a Piecewise Graph

The range is the union of all output values attained by all pieces.

Different pieces may overlap vertically, so range intervals should be merged when appropriate.

A Hole Does Not Always Remove a Range Value

Removing one point \((a,b)\) does not necessarily remove \(b\) from the range. Another input may still produce the same output \(b\).

Domain and range must be checked independently.

A Hole Always Removes That Specific Ordered Pair

Even when its \(x\)- or \(y\)-coordinate appears elsewhere, the open point itself is not part of the relation.

Intercepts Must Respect Piece Conditions

Solving one formula for an intercept is not enough. The resulting input must satisfy the condition attached to that piece.

A candidate zero outside the piece's domain is not an \(x\)-intercept of the piecewise function.

Example: Rejecting an Invalid Piecewise Zero

Suppose

\[ f(x)= \begin{cases} x-5, & x<2,\\ x+1, & x\ge2. \end{cases} \]

The first formula has algebraic zero \(x=5\), but \(5\not<2\). Therefore it contributes no zero to the piecewise function.

The \(y\)-Intercept Uses the Piece Containing \(x=0\)

To find a \(y\)-intercept, first determine which condition includes \(x=0\). Then evaluate only that active rule.

If \(0\) is outside the domain, there is no \(y\)-intercept.

Intersections with Other Graphs Must Respect Restrictions

If a piecewise graph intersects another relation, every candidate intersection must lie on an active portion of the appropriate piece.

Solving equations without checking interval restrictions can create false intersections.

Increasing and Decreasing Behavior Stops at Breaks

Monotonic intervals cannot simply pass across a discontinuity as though the graph were connected.

Analyze each continuous or discrete piece on its own domain interval before combining conclusions.

A Jump Can Change Global Order

A graph may rise on each separate piece but jump downward between them. It is therefore not necessarily increasing across the combined domain in the strict global sense.

Extreme Values Can Occur at Piece Boundaries

Closed endpoints and isolated points can be local or absolute extrema.

Open endpoints cannot attain an extreme value because they are not part of the graph.

A Supremum Need Not Be a Maximum

Suppose a piece approaches the height \(5\) at an open endpoint but never reaches it. The range may extend arbitrarily close to \(5\) without containing \(5\).

In that case, \(5\) is an upper bound but not a maximum value.

Piecewise Graphs Can Preserve Symmetry

A piecewise function may be even or odd if its pieces are arranged so the completed graph has the required symmetry.

Symmetry must be checked on the complete graph and domain, not on one formula alone.

Example: Absolute Value Revisited

The piecewise definition of \(|x|\) uses different formulas on negative and nonnegative inputs, yet the completed graph is symmetric about the \(y\)-axis.

The function is therefore even.

Transforming Piecewise Graphs

The entire graph can be translated, reflected, stretched, or compressed just like any other graph.

Endpoint locations and discontinuities move with the transformation.

Horizontal Transformations Also Change Conditions

When a piecewise formula is transformed horizontally, the interval conditions must be transformed as well.

Moving the formulas without moving their domain boundaries describes the wrong graph.

Example: Horizontal Translation of a Boundary

If a parent piece changes rule at \(x=2\), translating the entire graph right by \(5\) moves that boundary to

\[ x=7. \]

Open or closed status is preserved under the translation.

Vertical Transformations Preserve Input Conditions

A vertical translation or scaling changes outputs but does not change which inputs belong to each piece.

Thus the piece boundaries in \(x\) remain fixed under purely vertical transformations.

A Removable Discontinuity Moves with the Graph

If a graph has a hole at \((a,b)\), a translation by \((h,k)\) moves the hole to

\[ (a+h,b+k). \]

Its open status remains open unless the function definition is changed.

Discontinuity Type Is Often Preserved by Basic Transformations

Ordinary translations, reflections, and nonzero stretches or compressions do not repair a hole or merge the two sides of a jump. They move or rescale the feature.

Piecewise Models in Applications

Many real systems change rules when a threshold is crossed. Examples include tiered pricing, shipping fees, tax brackets, parking charges, overtime pay, control systems, and physical models with different operating regimes.

Threshold Models

A threshold is an input at which the applicable rule changes.

Whether the model is continuous at that threshold depends on the outputs of the neighboring rules, not merely on the fact that a threshold exists.

Example: Flat Fee Plus Usage

A service might charge

\[ C(x)= \begin{cases} 20, & 0\le x\le100,\\ 20+0.10(x-100), & x>100. \end{cases} \]

At \(x=100\), the second formula would also produce \(20\), so the pricing model joins continuously at the threshold.

Example: A Threshold Fee That Creates a Jump

If instead the second tier were

\[ C(x)=30+0.10(x-100) \quad\text{for }x>100, \]

then crossing the threshold would create an immediate jump of approximately \(10\) in cost.

Context Determines Whether a Jump Is Reasonable

A discontinuity in a mathematical model is not automatically an error. Some systems genuinely change by a fixed amount when a threshold is crossed.

The graph should reflect the rules of the situation faithfully.

Piecewise Linear Models

A piecewise linear function is made from line segments or rays. Such models are common when rates remain constant over ranges but change at selected breakpoints.

Breakpoints

A breakpoint is an input where the defining rule changes.

A breakpoint may be continuous, may form a corner, or may coincide with a discontinuity. The term itself does not specify which occurs.

Interpolation Across a Break Can Be Misleading

If a graph has a jump, connecting the two sides with a line segment invents outputs that the relation does not contain.

Never interpolate across a discontinuity unless the model explicitly tells you to do so.

Graphing Technology and Piecewise Syntax

Graphing software often uses logical restrictions or brace notation to limit a formula to part of its domain.

Syntax varies by platform, so the mathematical condition should be understood independently of the software command used to enter it.

Technology May Hide Endpoint Status

On a small screen, open and closed endpoints can be difficult to distinguish. A thick curve may also make a small hole nearly invisible.

Use the formula and conditions to verify what the display appears to show.

A Pixel Is Not a Mathematical Point

Digital graphs approximate curves with pixels. Apparent contact between two pieces does not prove exact equality at a boundary.

When exact continuity matters, evaluate the relevant formulas algebraically.

Window Choice Can Hide a Jump

A jump that is small relative to the vertical scale may appear connected. Zooming in can reveal the separation.

Conversely, an extremely narrow gap may disappear visually even though it is mathematically real.

A Vertical Asymptote Can Be Clipped

When outputs exceed the graphing window, software may simply stop drawing the curve. That clipping should not be mistaken for an ordinary endpoint.

The next lesson develops tools for distinguishing asymptotic behavior from finite endpoints and holes.

Tables Can Reveal Piecewise Behavior

A table with inputs just below and just above a breakpoint can help determine how neighboring pieces behave.

Use values close to the boundary rather than only widely separated sample points.

Example: Sampling Around a Boundary

To inspect behavior near \(x=2\), useful inputs might include

\[ 1.9,\quad1.99,\quad2,\quad2.01,\quad2.1. \]

The pattern of outputs can expose a hole, jump, or rapid unbounded behavior.

Do Not Confuse a Sparse Table with a Discontinuous Function

A table lists selected samples. Missing rows do not imply missing points on the graph unless the domain itself is discrete or restricted.

Continuous Versus Discrete Representation

A formula defined for every real input in an interval usually represents a continuous collection of inputs, even if only several sample points were used to construct the graph.

A discrete domain should be shown with separate points rather than an automatically connected curve.

Classifying a Graphical Break

When a graph breaks at \(x=c\), ask:

Do both sides approach the same finite height?
Do the two sides approach different finite heights?
Does at least one side grow without bound?
Is the issue simply that the domain ends or contains a larger gap?

These questions separate removable, jump, infinite, and domain-boundary behavior.

Classification: Same Finite Height

If both sides approach the same finite height but the point is missing or misplaced, the discontinuity is removable.

Classification: Different Finite Heights

If the left and right sides approach different finite heights, the graph has a jump discontinuity.

Classification: Unbounded Magnitude

If graph values increase or decrease without bound near the input, the graph has infinite discontinuity behavior, typically involving a vertical asymptote.

Oscillatory Discontinuity

Some advanced examples oscillate indefinitely near a point instead of approaching a finite value or growing monotonically without bound.

For instance, \(\sin(1/x)\) oscillates increasingly rapidly as \(x\) approaches \(0\). This is another way continuity can fail.

A Function Can Have Many Discontinuities

A function can have many discontinuities, even infinitely many. Step functions such as \(\lfloor x\rfloor\) have a jump at each integer.

Graphical methods help identify the pattern before more advanced analysis formalizes it.

Continuity on an Interval

Saying a function is continuous on an interval means that it is continuous at all relevant points of that interval, with one-sided interpretation at included endpoints when appropriate.

Intervals of Continuity

When discontinuities occur at isolated inputs, the domain can often be divided into intervals on which the graph is continuous.

A vertical asymptote at \(x=2\), for example, may separate continuity intervals to the left and right of \(2\).

Graphical Continuity Is Local

A graph can be continuous on one part of its domain and discontinuous on another.

Avoid labeling an entire complicated graph simply “continuous” or “discontinuous” when a more precise interval description is possible.

Exact Versus Estimated Breakpoints

If a formula gives the breakpoint exactly, report that exact value. If the breakpoint is read only from a graph, an estimate may be appropriate.

Do not convert an approximate graphical reading into an unjustified exact fraction or decimal.

A Reliable Procedure: Graphing Piecewise Definitions

List the formulas and their conditions.
Mark all boundary inputs on the \(x\)-axis.
Graph each formula only on its assigned interval or set.
Compute boundary heights.
Use open points for excluded boundaries.
Use closed points for included boundaries.
Check for gaps, jumps, holes, or continuous joins.
Verify that each input has the intended number of outputs.

A Reliable Procedure: Reading Piecewise Graphs

Locate the visible pieces.
Find their boundary inputs.
Read open and closed endpoints carefully.
Identify the equation or family of each piece when required.
Write the input restriction for each piece.
Combine the restrictions into a piecewise definition.

A Reliable Procedure: Checking Continuity at a Boundary

Determine the height approached from the left, if that side exists.
Determine the height approached from the right, if that side exists.
Compare the two approached heights.
Find the actual function value at the boundary, if one exists.
Decide whether all required values agree.

A Reliable Procedure: Classifying Discontinuities

Check whether the graph approaches one finite height.
If yes but the point is missing or misplaced, classify it as removable.
If the two sides approach different finite heights, classify it as a jump.
If values become unbounded, identify infinite discontinuity behavior.
If the graph oscillates without settling, note oscillatory behavior.
Distinguish these from ordinary domain endpoints and extended gaps.

Common Error: Graphing Every Formula Everywhere

Piecewise formulas are restricted. Drawing each formula across the entire coordinate plane creates extra points that are not part of the function.

Common Error: Ignoring Open and Closed Endpoints

Endpoint markers determine whether a boundary input is included. Reversing an open and closed point can change the function value, domain, range, and continuity classification.

Common Error: Assuming the Closed Point Must Be on the Upper Piece

Open and closed status comes from the inequality symbols, not from which point appears higher or lower on the graph.

Common Error: Treating a Formula Change as a Discontinuity

Two formulas can meet at exactly the same boundary point. Always compare their boundary behavior before declaring a break.

Common Error: Calling Every Sharp Point a Discontinuity

Corners and cusps can be continuous. A sudden change in direction is not the same as a break in the graph.

Common Error: Calling Every Missing Input a Hole

A hole is an isolated excluded point in an otherwise approaching graph. An entire missing interval is a different domain structure.

Common Error: Forgetting the Original Restriction After Cancellation

Cancelling a common factor can simplify a rational expression, but it does not restore an input that made the original denominator zero.

Common Error: Using an Inactive Piece to Find a Zero

Any solution produced by a piece's formula must also satisfy that piece's condition. Otherwise it is not on the piecewise graph.

Common Error: Connecting Discrete Pieces

Separate points should remain separate unless the domain includes the inputs between them. Connecting them invents additional ordered pairs.

Common Error: Trusting the Graphing Window

A clipped curve can resemble an endpoint, and a tiny jump can resemble a continuous join. Use the formulas, conditions, and scale to verify the display.

Common Error: Assuming an Undefined Point Has No Nearby Behavior

A function can be undefined at \(x=c\) while nearby graph values still approach a clear finite height. That is exactly the setting of a simple removable hole.

Common Error: Confusing Function Value with Approached Value

A filled point gives \(f(c)\). A nearby curve may approach a different height. Both pieces of information must be read separately.

Checking a Piecewise Graph Algebraically

Choose sample inputs from the interior of each piece and evaluate the assigned formula. Then test every boundary input according to its inclusion condition.

The plotted points should agree with those calculations.

Checking Boundary Consistency

At each breakpoint, evaluate the neighboring formulas at the boundary input even when one side excludes it. These values reveal whether the pieces would meet and whether a jump is present.

Checking with the Vertical-Line Test

After assembling all pieces, apply the vertical-line test. If a vertical line meets the completed graph more than once, the relation is not a function.

Checking Domain Coverage

Compare the union of the piece conditions with the intended domain. This catches accidental overlaps and unintended gaps.

Piecewise Notation as a Modeling Toolbox

Piecewise definitions let one mathematical object adapt to different input regimes without abandoning a unified function name.

They are especially powerful when a process changes behavior at known thresholds.

Breaks Are Features, Not Defects

A discontinuity may represent genuine structure: a threshold charge, a switch, a reset, a physical boundary, a forbidden input, or a change in operating regime.

The goal is to describe the break accurately, not to assume every graph should be smooth.

Piecewise Graphs Connect Earlier Graphical Ideas

A complete analysis may use many tools already developed in this section:

coordinate plotting;
relations and function tests;
intercepts and intersections;
domain and range;
increasing and decreasing behavior;
extreme values;
symmetry and family recognition;
transformations.

Practice 1: Evaluate a Piecewise Function

Let

\[ f(x)= \begin{cases} x+4, & x<1,\\ x^2, & x\ge1. \end{cases} \]

Find \(f(-2)\), \(f(1)\), and \(f(3)\).

Practice 2: Endpoint Status

For

\[ g(x)= \begin{cases} 2x-1, & x\le3,\\ 7-x, & x>3, \end{cases} \]

determine the open and closed boundary points at \(x=3\).

Practice 3: Continuous or Jump?

Determine whether the pieces meet continuously at \(x=2\):

\[ h(x)= \begin{cases} 3x-1, & x<2,\\ x+3, & x\ge2. \end{cases} \]

Practice 4: Removable Discontinuity

Identify the hole in

\[ r(x)=\frac{x^2-9}{x-3}. \]

Practice 5: Reject an Inactive Zero

Find all zeros of

\[ p(x)= \begin{cases} x-4, & x<0,\\ x+2, & x\ge0. \end{cases} \]

Practice 6: Domain from Conditions

Find the domain of

\[ q(x)= \begin{cases} x^2, & -3\le x<1,\\ \sqrt{x-2}, & x\ge2. \end{cases} \]

Practice 7: Classify the Discontinuity

A graph approaches \(5\) from the left of \(x=1\), approaches \(-2\) from the right, and has a filled point at \((1,5)\). Classify the discontinuity.

Practice 8: Step Function

For \(f(x)=\lfloor x\rfloor\), find \(f(2.8)\), \(f(-1.2)\), and \(f(4)\).

Practice 9: Build a Piecewise Definition

A graph follows \(y=x+1\) for inputs up to and including \(x=2\), and follows \(y=6-x\) for inputs greater than \(2\). Write a piecewise definition.

Practice 10: Identify the Error

A student graphs

\[ f(x)= \begin{cases} x+2, & x<1,\\ 4-x, & x\ge1 \end{cases} \]

by drawing both complete lines and says their intersection determines \(f(1)\). Explain the error and determine the actual value of \(f(1)\).

Practice Answers

Practice 1

Use the condition that contains each input:

\[ f(-2)=-2+4=\boxed{2}, \] \[ f(1)=1^2=\boxed{1}, \] \[ f(3)=3^2=\boxed{9}. \]

Practice 2

The first piece includes \(x=3\):

\[ 2(3)-1=5, \]

so \((3,5)\) is closed. The second piece excludes \(x=3\):

\[ 7-3=4, \]

so \((3,4)\) is open.

Practice 3

At the boundary, the left formula approaches

\[ 3(2)-1=5, \]

and the right formula gives

\[ 2+3=5. \]

The pieces meet at the same height, so the function is \(\boxed{\text{continuous at }x=2}\).

Practice 4

Factor:

\[ \frac{x^2-9}{x-3} =\frac{(x-3)(x+3)}{x-3} =x+3 \quad(x\ne3). \]

At \(x=3\), the corresponding height is \(6\). The hole is

\[ \boxed{(3,6)}. \]

Practice 5

The first formula has candidate zero \(x=4\), but \(4\not<0\), so reject it.

The second formula has candidate zero \(x=-2\), but \(-2\not\ge0\), so reject it.

Therefore the piecewise function has \(\boxed{\text{no zeros}}\).

Practice 6

The first piece contributes

\[ [-3,1), \]

and the second contributes

\[ [2,\infty). \]

Thus the domain is

\[ \boxed{[-3,1)\cup[2,\infty)}. \]

Practice 7

The two sides approach different finite heights, \(5\) and \(-2\). Therefore the graph has a

\[ \boxed{\text{jump discontinuity at }x=1}. \]

The fact that \(f(1)=5\) does not remove the jump.

Practice 8

\[ \lfloor2.8\rfloor=\boxed{2}, \qquad \lfloor-1.2\rfloor=\boxed{-2}, \qquad \lfloor4\rfloor=\boxed{4}. \]

Practice 9

\[ \boxed{ f(x)= \begin{cases} x+1, & x\le2,\\ 6-x, & x>2. \end{cases} } \]

Practice 10

The error is drawing each formula outside its assigned domain. At \(x=1\), only the second rule is active because its condition is \(x\ge1\).

\[ f(1)=4-1=\boxed{3}. \]

The intersection of the two unrestricted parent lines is irrelevant unless it lies on both restricted pieces.

What to Remember

A piecewise definition uses different rules on different input regions.
The condition attached to each formula is part of the definition.
Graph only the portion of each rule allowed by its condition.
Open points exclude boundaries; closed points include them.
A change of formula does not automatically create a discontinuity.
A removable discontinuity usually appears as a hole.
A jump occurs when the two sides approach different finite heights.
Infinite discontinuity involves unbounded behavior near an input.
Corners and cusps may still be continuous.
Zeros, intercepts, extrema, domain, and range must respect each piece's restriction.

The key habit is to separate three questions:

Which rule applies?
Is the boundary point included?
How do the neighboring pieces behave as they approach the boundary?

Where We Go Next

Piecewise graphs show how functions can break, jump, or change rules. Some graphs also approach lines or other simple structures without ever reaching them, and their behavior far from the origin can reveal the dominant algebraic structure of the function.

The next lesson, Asymptotes and End Behavior, develops methods for recognizing vertical, horizontal, and slant asymptotes and for describing how graphs behave as inputs become very large in magnitude or approach excluded values.