The Central Idea
Suppose
\[ f(x)=2x+1. \]As \(x\) increases, the output also increases:
\[ x=1\Rightarrow f(x)=3, \] \[ x=2\Rightarrow f(x)=5, \] \[ x=3\Rightarrow f(x)=7. \]This function is increasing.
Now consider
\[ g(x)=-3x+4. \]As \(x\) increases, the outputs decrease.
Finally, for
\[ h(x)=5, \]the output never changes.
These are the three basic behaviors:
- increasing;
- decreasing;
- constant.
Increasing Behavior
A function is increasing on an interval when larger inputs correspond to larger outputs.
More precisely, if
\[ x_1Decreasing Behavior
A function is decreasing on an interval when larger inputs correspond to smaller outputs.
If
\[ x_1Constant Behavior
A function is constant on an interval when every input in that interval gives the same output.
If
\[ x_1Read Behavior from Left to Right
When examining a graph, we usually move from left to right because the input variable increases in that direction.
As we move left to right:
- if the graph rises, the function is increasing;
- if the graph falls, the function is decreasing;
- if the graph stays horizontal, the function is constant.
Increasing Does Not Mean Positive
A function can be increasing while its outputs are negative.
For example,
\[ f(x)=x-10 \]is increasing everywhere.
At
\[ x=2, \]we have
\[ f(2)=-8, \]and at
\[ x=3, \]we have
\[ f(3)=-7. \]The outputs are negative, but they are increasing.
Decreasing Does Not Mean Negative
Consider
\[ f(x)=10-x. \]For many inputs, the outputs are positive.
But as \(x\) increases, \(f(x)\) decreases.
Sign and direction of change are different ideas.
Behavior Is About Change
The words increasing and decreasing describe how outputs compare as inputs change.
They do not describe whether the function values themselves are positive or negative.
Behavior Can Change
A function does not need to behave the same way across its entire domain.
Consider
\[ f(x)=x^2. \]As \(x\) moves from large negative values toward zero, the outputs decrease.
After \(x=0\), the outputs increase.
So the function is decreasing on one interval and increasing on another.
Example: \(f(x)=x^2\)
For
\[ f(x)=x^2, \]the function is decreasing on
\[ (-\infty,0) \]and increasing on
\[ (0,\infty). \]The point
\[ x=0 \]marks a change in behavior.
Intervals Matter
Statements about increasing or decreasing behavior should usually name an interval.
Instead of saying
“\(x^2\) is increasing,”
say
\[ f(x)=x^2 \]is increasing on
\[ (0,\infty). \]This is more precise because the function is not increasing on all real numbers.
Behavior from a Table
| \(x\) | \(f(x)\) |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 11 |
| 4 | 16 |
As the inputs increase, the outputs increase.
The table shows increasing behavior over the listed inputs.
A Decreasing Table
| \(x\) | \(f(x)\) |
|---|---|
| 0 | 20 |
| 1 | 17 |
| 2 | 14 |
| 3 | 11 |
As \(x\) increases, the output decreases.
A Constant Table
| \(x\) | \(f(x)\) |
|---|---|
| \(-2\) | 6 |
| 0 | 6 |
| 3 | 6 |
| 8 | 6 |
The output remains equal to \(6\).
The function is constant over the listed inputs.
Tables Show Sampled Behavior
A table may contain only selected inputs.
If the outputs increase across the listed rows, that establishes increasing behavior among those sampled values.
It does not always prove that the function increases at every point between them unless additional information about the function is known.
Strictly Increasing
The condition
\[ x_1Every increase in input produces a genuine increase in output.
Strictly Decreasing
The condition
\[ x_1is sometimes called strictly decreasing.
Every increase in input produces a genuine decrease in output.
Nondecreasing Behavior
A function is nondecreasing when
\[ x_1Nonincreasing Behavior
A function is nonincreasing when
\[ x_1Why the Distinction Matters
A function with flat intervals may be nondecreasing without being strictly increasing.
For example, a staircase-like function can remain constant for a while and then rise.
It never decreases, but not every larger input produces a larger output.
Constant Functions Are Both Nondecreasing and Nonincreasing
If
\[ f(x)=c, \]then for any
\[ x_1Therefore,
\[ f(x_1)\le f(x_2) \]and
\[ f(x_1)\ge f(x_2). \]So a constant function satisfies both weak monotonicity conditions.
Monotonicity
Monotonicity is the general idea of a function maintaining one directional order.
A function may be:
- increasing;
- decreasing;
- nondecreasing;
- nonincreasing;
- or monotonic only on certain intervals.
Linear Functions
For
\[ f(x)=mx+b, \]the parameter \(m\) determines the behavior:
- if \(m>0\), the function is increasing;
- if \(m<0\), the function is decreasing;
- if \(m=0\), the function is constant.
Why Slope Determines Linear Behavior
For two inputs
\[ x_1Because
\[ x_2-x_1>0, \]the sign of the output change is determined by the sign of \(m\).
Behavior and Average Rate of Change
Between inputs \(a\) and \(b\), with \(a\ne b\), the average rate of change is
\[ \frac{f(b)-f(a)}{b-a}. \]If
\[ a- a positive average rate means the net output increased;
- a negative average rate means the net output decreased;
- a zero average rate means the two endpoint outputs are equal.
Endpoint Change Does Not Prove Behavior Between Them
Suppose
\[ f(a)But the function could have risen, fallen, and risen again between \(a\) and \(b\).
Net change and continuous increasing behavior are not the same claim.
Example of Net Increase Without Always Increasing
A function might satisfy
\[ f(0)=2 \]and
\[ f(10)=8, \]yet fall to \(1\) somewhere between those inputs.
The overall change is positive, but the function was not increasing across the entire interval.
Turning Points
A turning point occurs when a graph changes from increasing to decreasing or from decreasing to increasing.
Such points often correspond to local high or low values.
Local Maximum Preview
If a function increases and then begins decreasing, the transition may form a local maximum.
The function is locally higher there than at nearby inputs.
Local Minimum Preview
If a function decreases and then begins increasing, the transition may form a local minimum.
The function is locally lower there than at nearby inputs.
Example: A Parabola
Consider
\[ f(x)=(x-3)^2+2. \]The function decreases as \(x\) approaches \(3\) from the left and increases after \(x=3\).
So it is decreasing on
\[ (-\infty,3) \]and increasing on
\[ (3,\infty). \]The point
\[ (3,2) \]is a minimum.
Constant Intervals
A function may be constant only on part of its domain.
For example, a piecewise function could rise, remain flat, and then rise again.
The flat portion is a constant interval.
Piecewise Example
Consider
\[ f(x)= \begin{cases} x+2, & x<0,\\ 2, & 0\le x\le3,\\ x-1, & x>3. \end{cases} \]The function is:
- increasing for \(x<0\);
- constant on \([0,3]\);
- increasing again for \(x>3\).
Behavior Need Not Be Continuous
A function can be increasing or decreasing even if its graph has jumps.
Monotonicity compares the order of outputs as inputs increase.
Continuity is a separate property.
Discrete Functions Can Increase Too
Consider the sequence function
\[ a(n)=2n+1, \qquad n\in\mathbb{Z}_{>0}. \]Then
\[ a(1)=3,\quad a(2)=5,\quad a(3)=7. \]The function is increasing on its discrete domain.
Increasing Sequences
For a sequence
\[ a_n, \]one common test for increasing behavior is
\[ a_{n+1}>a_n. \]For nondecreasing behavior, use
\[ a_{n+1}\ge a_n. \]Decreasing Sequences
A sequence is decreasing when
\[ a_{n+1}Example: Geometric Sequence
Consider
\[ a_n=2^n. \]Each term is twice the previous term, so the sequence is increasing.
By contrast,
\[ b_n=\left(\frac12\right)^n \]is decreasing for positive integer \(n\).
Increasing Can Be Fast or Slow
Both
\[ f(x)=x \]and
\[ g(x)=x^3 \]are increasing on all real numbers.
But they do not increase at the same rate.
Monotonicity describes direction, not speed of change.
Constant Rate Is Different from Constant Function
A linear function such as
\[ f(x)=3x+2 \]has a constant rate of change, but it is not a constant function.
The output changes steadily.
A constant function such as
\[ g(x)=5 \]has output that does not change at all.
Zero Rate of Change and Constant Behavior
For a constant function, the average rate of change between any two distinct inputs is
\[ 0. \]This is because
\[ f(b)-f(a)=0. \]Increasing and One-to-One Behavior
A strictly increasing function cannot assign the same output to two different inputs.
If
\[ x_1Thus, a strictly increasing function is one-to-one on that interval.
Decreasing and One-to-One Behavior
A strictly decreasing function is also one-to-one on the interval where it decreases.
Different ordered inputs produce different ordered outputs.
Why Monotonicity Helps with Inverses
If a function is strictly increasing or strictly decreasing on an interval, it passes the horizontal-line test there.
This often allows an inverse function to be defined on that interval.
Monotonicity therefore connects directly to invertibility.
Global Versus Local Behavior
A function may be:
- globally increasing on its whole domain;
- increasing only on a particular interval;
- or increasing near one point but decreasing elsewhere.
Always pay attention to the interval under discussion.
Intervals of Increase and Decrease
When reporting graph behavior, we often list intervals.
For example:
\[ \text{decreasing on }(-\infty,2), \] \[ \text{increasing on }(2,\infty). \]Open intervals are commonly used around turning points when describing where the function is actively increasing or decreasing.
Why Turning Points Are Often Excluded
At the exact turning point, the function is not moving “through” an interval in either direction.
So behavior intervals are often written on either side of the point.
The point itself still belongs to the function's domain unless otherwise restricted.
Different Textbooks Use Slightly Different Conventions
Some texts use increasing to mean strictly increasing.
Others use it more loosely for nondecreasing behavior.
Malone Math will distinguish them when the difference matters:
- strictly increasing: \(f(x_1)
- nondecreasing: \(f(x_1)\le f(x_2)\).
Behavior and Derivatives Preview
Later in calculus, the derivative gives a local rate of change.
Very roughly:
- \(f'(x)>0\) suggests increasing behavior;
- \(f'(x)<0\) suggests decreasing behavior;
- \(f'(x)=0\) may indicate a flat point or turning point.
This is a future refinement of the ideas introduced here.
A Zero Derivative Does Not Automatically Mean Constant
A single point where
\[ f'(x)=0 \]does not mean the entire function is constant.
For example,
\[ f(x)=x^2 \]has a flat tangent at \(x=0\), yet the function decreases before that point and increases after it.
Behavior in Context
Suppose
\[ T(t) \]represents temperature over time.
If \(T\) is increasing on an interval, the temperature is rising during that time.
If \(T\) is decreasing, it is cooling.
If \(T\) is constant, the temperature remains unchanged.
Population Example
If
\[ P(t) \]represents population, then increasing behavior means later times correspond to larger modeled populations.
Decreasing behavior means the population is falling.
Constant behavior means the population remains unchanged over the interval.
Cost Example
Suppose
\[ C(q) \]is total cost as production quantity \(q\) changes.
An increasing cost function means greater production corresponds to greater total cost.
This does not tell us whether the cost rises slowly or quickly; that requires rate-of-change information.
Worked Example: Linear Function
Let
\[ f(x)=5x-2. \]The slope is
\[ 5>0. \]Therefore, \(f\) is increasing on
\[ \boxed{(-\infty,\infty)}. \]Worked Example: Decreasing Linear Function
Let
\[ f(x)=-4x+9. \]The slope is
\[ -4<0. \]Therefore, \(f\) is decreasing on
\[ \boxed{(-\infty,\infty)}. \]Worked Example: Constant Function
Let
\[ f(x)=7. \]Every input gives output \(7\).
Therefore, \(f\) is constant on its entire domain.
Worked Example: Quadratic Behavior
Let
\[ f(x)=(x+1)^2-4. \]The vertex occurs at
\[ x=-1. \]The function decreases as \(x\) approaches \(-1\) from the left and increases afterward.
So it is:
\[ \boxed{\text{decreasing on }(-\infty,-1)} \]and
\[ \boxed{\text{increasing on }(-1,\infty)}. \]Worked Example: Table Behavior
| \(x\) | \(f(x)\) |
|---|---|
| 0 | 10 |
| 1 | 8 |
| 2 | 8 |
| 3 | 11 |
Across the sampled values:
- the output decreases from \(x=0\) to \(x=1\);
- stays constant from \(x=1\) to \(x=2\);
- increases from \(x=2\) to \(x=3\).
Worked Example: Average Rate of Change
Suppose
\[ f(2)=5 \]and
\[ f(6)=17. \]The average rate of change from \(2\) to \(6\) is
\[ \frac{17-5}{6-2} = \frac{12}{4} = 3. \]The positive result tells us the net change over the interval is upward.
It does not by itself prove the function increased at every point between \(2\) and \(6\).
A Reliable Graph-Behavior Workflow
- Read the graph from left to right.
- Mark where the graph rises, falls, or stays level.
- Locate turning points and flat intervals.
- Write the corresponding input intervals.
- Distinguish strict increase/decrease from nondecreasing/nonincreasing behavior when necessary.
- Check domain gaps before joining intervals.
A Reliable Table-Behavior Workflow
- Order the inputs from least to greatest.
- Compare consecutive outputs.
- Record whether each change is upward, downward, or zero.
- Remember that a finite table may show only sampled behavior.
- Use a formula or graph when stronger interval claims are needed.
A Reliable Algebraic Workflow
- Identify the function family when possible.
- Use known structural information such as slope or vertex.
- Compare \(f(x_1)\) and \(f(x_2)\) when a direct proof is useful.
- Break the domain into intervals if behavior changes.
- Check any domain restrictions.
- State the behavior with explicit intervals.
Common Errors
Confusing Increasing with Positive
An increasing function can have negative outputs.
Confusing Decreasing with Negative
A decreasing function can remain entirely above zero.
Describing a Function Globally When Behavior Changes
State the intervals on which each behavior occurs.
Using Endpoint Change to Prove Monotonicity
A positive net change does not guarantee that the function increased throughout the interval.
Confusing Constant Function with Constant Rate of Change
A nonhorizontal line has constant rate of change but is not a constant function.
Ignoring Flat Intervals
A function may be nondecreasing without being strictly increasing.
Assuming Continuity Is Required
Monotonic functions can contain jumps.
Practice
1. Linear Behavior
Determine the behavior of
\[ f(x)=7x-4. \]Check Answer
The slope is positive, so
\[ \boxed{f\text{ is increasing on }(-\infty,\infty)}. \]2. Decreasing Linear Function
Determine the behavior of
\[ g(x)=-2x+5. \]Check Answer
The slope is negative, so
\[ \boxed{g\text{ is decreasing on }(-\infty,\infty)}. \]3. Constant Function
Determine the behavior of
\[ h(x)=12. \]Check Answer
The output never changes, so \(h\) is constant on its domain.
4. Quadratic Behavior
For
\[ f(x)=(x-4)^2, \]state the intervals of decrease and increase.
Check Answer
\[ \boxed{\text{decreasing on }(-\infty,4)} \]and
\[ \boxed{\text{increasing on }(4,\infty)}. \]5. Positive or Increasing?
The function
\[ f(x)=x-100 \]has negative values for many inputs. Is it increasing or decreasing?
Check Answer
It is increasing because its slope is
\[ 1>0. \]6. Sampled Table Behavior
| \(x\) | \(f(x)\) |
|---|---|
| 1 | 9 |
| 2 | 7 |
| 3 | 7 |
| 4 | 10 |
Describe the changes across the listed values.
Check Answer
The outputs decrease from \(1\) to \(2\), remain constant from \(2\) to \(3\), and increase from \(3\) to \(4\).
7. Strictly Increasing or Nondecreasing?
A function rises, stays flat on one interval, and then rises again without ever falling. Is it necessarily strictly increasing?
Check Answer
No. It is nondecreasing, but the flat interval prevents it from being strictly increasing across the whole domain.
8. Average Rate of Change
If
\[ f(1)=4 \]and
\[ f(5)=12, \]find the average rate of change.
Check Answer
\[ \frac{12-4}{5-1} = \frac84 = \boxed{2}. \]Key Ideas
- A function is increasing when larger inputs produce larger outputs.
- A function is decreasing when larger inputs produce smaller outputs.
- A function is constant when the output remains unchanged.
- Increasing and decreasing describe direction of change, not whether outputs are positive or negative.
- Behavior should usually be stated on intervals.
- A function can change from increasing to decreasing or vice versa.
- Turning points often separate intervals of different behavior.
- Strictly increasing and strictly decreasing are stronger than nondecreasing and nonincreasing.
- Constant functions are both nondecreasing and nonincreasing.
- Linear behavior is determined by the sign of the slope.
- A positive average rate of change shows positive net change, not necessarily increasing behavior at every point.
- Monotonicity can be studied for continuous, discontinuous, and discrete functions.
- Strict monotonicity implies one-to-one behavior on the interval.
- These ideas prepare the way for extrema, inverse functions, rates of change, and derivative tests.
Where This Leads
Increasing, decreasing, and constant behavior provide the first language for describing how a function changes across its domain.
These ideas lead naturally into local and global extrema, average and instantaneous rates of change, monotonicity, inverse functions, optimization, derivative sign charts, and the broader study of how mathematical models evolve as their inputs vary.