Inverse Functions and Restricted Domains

The Central Idea

Consider

If

then the inverse should reverse that assignment:

The original function sends

The inverse sends

An inverse function reverses the arrows of a function.

What Is an Inverse Function?

If

is bijective, then its inverse is a function

such that

for every \(x\in A\), and

for every \(y\in B\).

The Inverse Reverses the Assignment

If

then

This is the defining reversal.

Inverse Notation Does Not Mean Reciprocal

The notation

does not mean

For example, if

then

while

These are different functions.

Why Bijectivity Matters

For a full inverse function from codomain back to domain, the original function must be bijective.

That means it must be both:

• injective;
• surjective.

Why Injectivity Is Necessary

Suppose two distinct inputs satisfy

If we reverse the assignment, the inverse would need to send

to both \(a\) and \(b\).

That would give one input two outputs, so the inverse relation would not be a function.

Why Surjectivity Is Necessary

Suppose the codomain contains an element \(y\) that is never produced by \(f\).

Then \(f^{-1}(y)\) would have no value.

So a full inverse defined on the entire original codomain requires surjectivity.

Inverse Relation Versus Inverse Function

Every relation can have its ordered pairs reversed.

If the original relation contains

the inverse relation contains

But the inverse relation is a function only when the reversal still assigns exactly one output to each input.

Ordered-Pair Example

Suppose

Then

This inverse relation is also a function because the original outputs were all distinct.

Noninvertible Ordered-Pair Example

Suppose

Then the inverse relation would contain

and

So the inverse relation is not a function.

Horizontal-Line Test

A real-valued function has an inverse function on its domain if it is one-to-one there.

Graphically, this is tested with the horizontal-line test.

If every horizontal line meets the graph at most once, the function is injective and can be reversed on its range.

Why the Horizontal-Line Test Works

A horizontal line corresponds to one output value.

If that line intersects the graph more than once, the same output came from several inputs.

Reversing those assignments would fail the function rule.

Domain and Range Swap

If

is bijective, then

Thus:

and

Graph of an Inverse

If

lies on the graph of \(f\), then

lies on the graph of \(f^{-1}\).

So the graphs of a function and its inverse are reflections across the line

Why Reflection Across \(y=x\) Appears

Reversing an ordered pair swaps its coordinates:

Geometrically, swapping coordinates reflects a point across the line

Finding an Inverse Algebraically

For a function given by a formula, a common workflow is:

1. write \(y=f(x)\);
2. swap \(x\) and \(y\);
3. solve for \(y\);
4. rename the result \(f^{-1}(x)\);
5. state the domain and range.

Linear Inverse Example

Let

Write

Swap \(x\) and \(y\):

Solve for \(y\):

Therefore,

Verify by Composition

For

and

check:

Also,

Both compositions return the identity.

Identity Function Connection

The identity function is

For inverse functions,

and

Why Composition Is the Strongest Check

Finding a formula that seems to reverse another function is not enough.

Composition verifies that the two functions actually undo each other on the correct domains.

The Problem with \(f(x)=x^2\)

Consider

on

The function is not injective because

and

So it has no inverse function on all of \(\mathbb{R}\).

Algebra Reveals the Problem

Write

Swap:

Then

The symbol

reveals two possible outputs for many inputs.

That is an inverse relation, not one inverse function.

Restricted Domains

To create an inverse function, we can restrict the original domain to a region where the function is one-to-one.

For

a common restriction is

Square Function with Restricted Domain

Define

This function is strictly increasing, injective, and surjective onto its codomain.

Therefore, it is bijective.

Inverse of the Restricted Square Function

Now the inverse is

The principal square root corresponds precisely to the nonnegative domain restriction on the original square function.

A Different Restriction Gives a Different Inverse

Instead, restrict

to

Then the inverse becomes

The chosen branch reflects the chosen original domain.

Restricted Domain Is Part of the Function

The formulas

and

are different functions because their domains differ.

They therefore have different inverses.

Strict Monotonicity and Inverses

A strictly increasing or strictly decreasing function is injective on that interval.

Therefore, strict monotonicity is a powerful way to identify intervals on which an inverse function can exist.

Quadratic Domain Restriction

Consider

The vertex occurs at

The function is decreasing on

and increasing on

Either interval can be used as a one-to-one restriction.

Inverse on the Right Branch

Restrict

to

Write

Then

Because \(x\ge3\),

So

Therefore,

Inverse on the Left Branch

If instead the original domain is

then

So the inverse is

The Original Range Becomes the Inverse Domain

For

on either one-to-one branch, the range is

Therefore, the inverse domain is

Absolute Value and Restricted Domains

The absolute-value function

is not injective on \(\mathbb{R}\) because

But on

it becomes

which is bijective onto

Exponential and Logarithmic Functions

The exponential function

is strictly monotonic and has range

Its inverse is the logarithmic function

Domain and Range Swap for Exponentials

For

we have

Therefore,

Cube and Cube Root

Consider

This function is strictly increasing on all real numbers and has range \(\mathbb{R}\).

Therefore, it is bijective from \(\mathbb{R}\) to \(\mathbb{R}\).

Its inverse is

Reciprocal Function

Consider

Its range also excludes zero.

If

then

implies

So

The reciprocal function is its own inverse.

Self-Inverse Functions

A function satisfying

is called self-inverse or an involution.

Such a function satisfies

Reflection Across \(y=x\) for Self-Inverse Functions

If a function is its own inverse, its graph is symmetric about

The reciprocal function is a familiar example.

Inverse of a Linear Function

For

solve

Then

So

Therefore,

Constant Functions Have No Inverse Function

If

then every input shares the same output.

Such a function is never injective when the domain has more than one element.

So it cannot have an inverse function on that domain.

Piecewise Functions Can Be Invertible

A piecewise function may have an inverse if the entire function is one-to-one and reaches its intended codomain.

Its inverse may itself require piecewise notation.

Piecewise Inverse Example

Suppose

The first branch has range

and the second has range

These ranges do not overlap, so the function is injective.

Inverse of the Piecewise Example

Reverse each branch:

and

Thus,

The inverse domain is the original range.

Inverse Functions and Composition

If \(f\) and \(g\) are invertible, then

The order reverses.

Why the Order Reverses

If the original process applies

first and then

the inverse process must undo \(g\) first and then undo \(f\).

This mirrors the reverse-order logic used in solving equations.

Inverse Functions and Solving Equations

Suppose

If \(f\) is invertible, apply \(f^{-1}\) to both sides:

Then

Inverse functions provide a general framework for undoing operations.

Inverse Operations and Inverse Functions

Solving

uses subtraction and division to undo the original steps.

The inverse function

packages those reverse operations into one function.

Inverse Functions Are Directional

The function

and its inverse

reverse roles:

• original inputs become inverse outputs;
• original outputs become inverse inputs.

Codomain Choice Still Matters

Suppose

The function is injective but not surjective onto \(\mathbb{R}\).

So it is not bijective under that codomain declaration.

Choose the Natural Codomain for Invertibility

Instead define

Now the range equals the codomain.

The function is bijective and has inverse

Inverse Function Versus Solving for \(x\)

Solving one equation such as

asks for preimages of one output.

Finding \(f^{-1}\) constructs a new function that solves the entire family of equations

at once.

Inverse Functions and Fixed Points

If

then \(a\) is a fixed point.

For an invertible function,

as well.

Fixed points remain fixed under both directions of the mapping.

Inverse Graphs Share Fixed Points

A function and its inverse are reflected across

Points lying on that reflection line remain unchanged.

So fixed points are shared by a function and its inverse.

Parametric View of Inverses

If a graph is parametrized by

then the inverse relation swaps the coordinates:

This gives another way to see the reflection across \(y=x\).

Worked Example: Find and Verify an Inverse

Let

Write

Swap:

Solve:

Thus,

Check:

Worked Example: Restricted Quadratic

Let

with domain

Then

Swap:

Solve:

Therefore,

The inverse domain is

Worked Example: Determine Whether an Inverse Exists

Let

on \(\mathbb{R}\).

Because

and

the function is not injective.

So it has no inverse function on all real numbers.

Worked Example: Choose a Restricted Domain

For

restrict to

Then

Swap:

So

Using the nonnegative branch,

The inverse domain is

A Reliable Inverse-Function Workflow

1. State the original domain and codomain.
2. Check injectivity using algebra, monotonicity, or the horizontal-line test.
3. Check whether the range matches the intended codomain.
4. If necessary, restrict the domain to a one-to-one interval.
5. Write \(y=f(x)\).
6. Swap \(x\) and \(y\).
7. Solve for \(y\).
8. Rename the result \(f^{-1}(x)\).
9. State the inverse domain and range.
10. Verify by composition.

A Reliable Domain-Restriction Workflow

1. Identify where the function fails to be one-to-one.
2. Use turning points or monotonic intervals to locate one-to-one branches.
3. Choose a branch appropriate to the problem or convention.
4. Record the restricted domain explicitly.
5. Determine the corresponding range.
6. Use that range as the domain of the inverse.

A Reliable Graph Workflow

1. Apply the horizontal-line test.
2. If the graph fails, identify a one-to-one branch.
3. Restrict the domain to that branch.
4. Reflect the restricted graph across \(y=x\).
5. Check that original domain and range have swapped.

Common Errors

Confusing \(f^{-1}(x)\) with \(1/f(x)\)

Inverse functions reverse mappings; reciprocals divide \(1\) by an output.

Finding an Algebraic Inverse Without Checking Injectivity

A formula that produces \(\pm\) branches signals that the original function may need a restricted domain.

Forgetting Domain and Range

The inverse domain is the original range, and the inverse range is the original domain.

Ignoring Codomain

A function may be injective but not bijective under an overly large codomain.

Choosing Both Square-Root Branches

An inverse function must return exactly one output. The original restricted domain determines which branch is correct.

Assuming Every Function Has an Inverse Function

Only one-to-one functions can be reversed as functions on their ranges.

Failing to Verify by Composition

The strongest check is that both compositions reduce to the identity on the appropriate domains.

Practice

1. Linear Inverse

Find the inverse of

2. Does an Inverse Exist?

Does

have an inverse function on \(\mathbb{R}\)?

3. Restrict the Square Function

Restrict

to \(x\ge0\). Find the inverse.

4. Domain and Range Swap

If

and

what are the domain and range of \(f^{-1}\)?

5. Composition Check

Let

and

Compute \(g(f(x))\).

6. Quadratic Restriction

Let

with domain \(x\ge2\). Find the inverse.

7. Reciprocal Function

What is the inverse of

8. Graphical Relationship

How are the graphs of \(f\) and \(f^{-1}\) related?

Key Ideas

• An inverse function reverses the input-output assignment of a function.
• If \(f(a)=b\), then \(f^{-1}(b)=a\).
• \(f^{-1}(x)\) is not the same as \(1/f(x)\).
• A full inverse function requires bijectivity between the stated domain and codomain.
• Injectivity prevents one inverse input from requiring several outputs.
• Surjectivity ensures every inverse input has an output.
• The domain and range of a function swap under inversion.
• The graphs of \(f\) and \(f^{-1}\) are reflections across \(y=x\).
• The horizontal-line test checks whether a real-valued function is one-to-one.
• A noninjective function can often be made invertible by restricting its domain to a one-to-one branch.
• Strictly monotonic functions are injective on their intervals of monotonicity.
• Quadratic inverses require choosing a branch consistent with the restricted original domain.
• Inverse functions satisfy \(f^{-1}\circ f=I\) and \(f\circ f^{-1}=I\).
• Inverse functions provide a general language for undoing operations and solving equations.

Where This Leads

Inverse functions complete the basic structural picture of functions by showing when an input-output relationship can be reversed without ambiguity.

These ideas lead directly into logarithms, inverse trigonometric functions, coordinate transformations, inverse matrices, solving equations with inverse operations, inverse modeling problems, implicit relations, change of variables, and higher mathematics where reversible mappings and structural equivalence are central.