Find and interpret inverse relations and restricted inverse functions

Alberta Grade 12 Mathematics

Relations and Functions

Reverse input and output, and restrict a domain when needed

A machine takes an input and produces an output. An inverse reverses that process: the original output becomes the new input, and the original input becomes the new output. If a machine doubles a number and adds three, its inverse subtracts three and then halves. Sometimes different inputs produce the same output. Reversing those pairings may give an inverse relation that is not a function. Restricting the original domain can sometimes make its inverse a function.

What you will learn

Find an inverse relation by reversing input and output.
Find an inverse function algebraically and state its domain and range.
Explain when a domain restriction is needed to make an inverse a function.
Interpret inverse relations and functions using equations, tables, and graphs.

1. Review: inputs, outputs, and pairings

A relation is a set of pairings between inputs and outputs. A function is a relation that gives exactly one output for each input. For a function written as

y=f(x)

,

x

is the input and

y

is the output.

The domain is the set of allowed inputs. The range is the set of outputs that result. For example, the rule

y=2x+3

sends input

4

to output

11

. We write

f(4)=11

.

A point

(a,b)

on a graph means input

a

is paired with output

b

. Reversing the pairing changes it to

(b,a)

. This coordinate switch is the central idea behind an inverse.

(a,b)∈ R\iff(b,a)∈ R^{-1}

A function gives exactly one output for each allowed input.
The domain lists allowed inputs; the range lists resulting outputs.
Reversing a pair changes $(a,b)$ to (b,a)

2. Inverse relations and inverse functions

The inverse relation reverses every ordered pair in the original relation. To find its equation, switch

x

and

y

. If possible, solve the new equation for

y

.

The notation

f^{-1}(x)

is used for an inverse function. The exponent

-1

does not mean a reciprocal. It means that the function sends an output of

f

back to the input that produced it. If

f(a)=b

, then

f^{-1}(b)=a

.

An inverse relation is not always a function. For example, squaring sends both

2

and

-2

to

4

. After reversing the pairs, input

4

has two outputs:

2

and

-2

.

A function is one-to-one if different inputs always produce different outputs. The horizontal line test checks this on a graph: if a horizontal line crosses the graph more than once, the function is not one-to-one on that domain.

A domain restriction selects only some of the original inputs. It can make a function one-to-one. For a function and its inverse, the original range becomes the inverse domain, and the original restricted domain becomes the inverse range.

(a,b)∈ R\Rightarrow(b,a)∈ R^{-1}

The inverse relation reverses pairs and switches domain and range.
An inverse is a function only if each original output comes from one input on the chosen domain.
A domain restriction can make a function one-to-one.

3. Finding, graphing, and interpreting inverses

To find an inverse algebraically, write the original rule as

y=f(x)

, switch

x

and

y

, and solve for

y

. The result describes the inverse relation. Check whether it gives one output for each allowed input before calling it an inverse function.

The graph of an inverse is the reflection of the original graph across the line

y=x

. Reflection across this line switches the coordinates of every point. The vertical line test checks whether a graph represents a function: if a vertical line crosses it more than once, it is not a function.

For a quadratic such as

y=x^2

, positive and negative inputs can produce the same output. Restricting the original domain to

x\geq 0

keeps the nonnegative inputs; restricting it to

x\leq 0

keeps the nonpositive inputs. Each choice leads to a different inverse function.

When finding the inverse of a restricted quadratic, choose the square-root sign that matches the restriction. State the restriction and the inverse's domain and range. Keep exact values unless an approximation is requested.

The original range becomes the inverse domain, and the original restricted domain becomes the inverse range.

\operatorname{Dom}(f^{-1})=\operatorname{Range}(f)

Switch variables and isolate the new $y$ to find an inverse rule.
Inverse graphs reflect across $y=x$ .
The original range becomes the inverse domain; the original restricted domain becomes the inverse range.

4. A useful routine

First identify the original domain and range, or determine them from the rule. Next reverse the input-output pairings by switching variables. Solve for the new output when possible. Then check whether the inverse is a function. If it is not, decide whether a suitable original domain restriction is part of the question. Finish by stating the inverse's domain and range.

A table can make the reversal visible before you work with an equation.

Check the function condition after reversing the pairs.
State restrictions and domain and range explicitly.
Use a table or graph to interpret what the inverse does.

A relation and its inverse

Original input	Original output	Reversed pair
2	7	$(7,2)$
4	9	$(9,4)$
6	11	$(11,6)$

Worked example

1. Invert a linear function

Find the inverse of

f(x)=2x-5

. State the domain and range of both functions, assuming the original domain is all real numbers.

A line and its inverse

The lines reflect across $y=x$. This window shows only part of their full domains.

Write the rule
Represent the output as $y$ . Switching the variables reverses the input-output pairs.
$y=2x-5$
Switch variables
The original input becomes the new output, and the original output becomes the new input.
$x=2y-5$
Solve for the new output
Add five to both sides, then divide by two. Every real input gives one real output, so the inverse is a function.
$y=\frac{x+5}{2}$
State domain and range
A non-constant linear function with no input restriction can produce every real output. Its inverse also has all real numbers as its domain and range.
$\operatorname{Dom}(f)=\operatorname{Range}(f)=\operatorname{Dom}(f^{-1})=\operatorname{Range}(f^{-1})=\mathbb{R}$

Answer:

f^{-1}(x)=\frac{x+5}{2}

. Both functions have domain

\mathbb{R}

and range

\mathbb{R}

.

Check: For example,

f(4)=3

, and the inverse sends

3

back to

4

:

f^{-1}(3)=4

.

Worked example

2. Restrict a quadratic to make an inverse function

Let

f(x)=(x-1)^2+2

with the restriction

x\geq 1

. Find the inverse function and state its domain and range.

Restricted quadratic and inverse

The original curve is defined only for $x\geq1$, and the inverse curve only for $x\geq2$. This window shows only part of each curve's domain.

Use the restriction
The quadratic normally has two inputs for many outputs. Keeping only $x\geq 1$ keeps the part to the right of its vertex, where each output comes from one input.
$x\geq 1$
Switch variables
Exchange the variables to reverse each input-output pair.
$x=(y-1)^2+2$
Solve for the inverse output
Subtract two and take a square root. The inverse outputs are original inputs, so they must satisfy $y\geq 1$ . Use the positive square root.
$y=1+\sqrt{x-2}$
Give domain and range
The original function has minimum value $2$ , so its range is $[2,\infty)$ . This becomes the inverse domain. The restricted original domain becomes the inverse range.
$\operatorname{Dom}(f^{-1})=[2,\infty),\quad\operatorname{Range}(f^{-1})=[1,\infty)$

Answer:

f^{-1}(x)=1+\sqrt{x-2}

for

x\geq 2

. Its range is

[1,\infty)

.

Check: At the vertex,

f(1)=2

. The inverse gives

f^{-1}(2)=1

, so input and output have been reversed.

Worked example

3. Find an inverse relation that is not a function

Find the inverse relation of

y=x^2-4

when the original domain is all real numbers. Explain why the inverse relation is not a function.

A relation and its inverse relation

The inverse has two branches and is not a function. This graph window shows only part of each relation.

Switch variables
Switching the variables reverses every ordered pair in the original relation.
$x=y^2-4$
Solve for the possible outputs
Add four and take square roots. Both signs are needed because the original domain includes positive and negative inputs.
$y=\pm\sqrt{x+4}$
Interpret the relation
The inverse relation has domain $x\geq -4$ . At the endpoint $x=-4$ , the two signs give the same output, $0$ . For every input greater than $-4$ , there are two different outputs, so the inverse relation is not a function.
$\operatorname{Dom}(f^{-1})=[-4,\infty)$
Connect to a restriction
If the original domain were restricted to $x\geq 0$ , only the positive branch would remain. A restriction to $x\leq 0$ keeps only the negative branch. These inverse functions are $y=\sqrt{x+4}$ and $y=-\sqrt{x+4}$ , respectively.

Answer: The inverse relation is

y=\pm\sqrt{x+4}

, with domain

x\geq -4

. At

x=-4

both signs give

0

; for every

x>-4

, the relation gives two different outputs. It is not a function.

Check: The original pairs

(2,0)

and

(-2,0)

reverse to

(0,2)

and

(0,-2)

. The inverse therefore assigns two outputs to input

0

.

Common mistakes and how to avoid them

Treating

f^{-1}(x)

as the reciprocal of

f(x)

.

Correction: The inverse reverses input and output. It is not found by taking one divided by the function value.

Switching variables but not solving for the new output.

Correction: After switching

x

and

y

, isolate

y

to write the inverse rule when possible.

Keeping both square-root signs after restricting the original domain.

Correction: Use the sign that matches the original domain restriction. Keeping both signs gives a relation that is not a function.

Giving the inverse the same domain and range as the original.

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

Lesson summary

An inverse relation reverses each ordered pair.
To find an inverse rule, switch $x$ and $y$ and solve for $y$ .
An inverse is a function only when the original function is one-to-one on its domain.
Restricting a domain can make a function one-to-one and produce an inverse function.
Inverse graphs reflect across $y=x$ , and their domains and ranges switch.

Check your understanding

Question 1

The function

f(x)=3x+1

has domain all real numbers. Which rule is its inverse?

$f^{-1}(x)=\frac{x-1}{3}$
$f^{-1}(x)=3x-1$
$f^{-1}(x)=\frac{x+1}{3}$
$f^{-1}(x)=\frac{1}{3x+1}$

Show answer and explanation

f^{-1}(x)=\frac{x-1}{3}

Switch variables to get

x=3y+1

, then solve to obtain

y=\frac{x-1}{3}

.

Question 2

For

f(x)=x^2

with domain

x\geq 0

, what is the domain of its inverse?

$[0,\infty)$
$(-\infty,\infty)$
$(-\infty,0]$
$[-1,1]$

Show answer and explanation

[0,\infty)

The original range is

[0,\infty)

, and that becomes the inverse domain.

Question 3

The relation

y=x^2

has domain all real numbers. Why is its inverse not a function?

Some inverse inputs have two possible outputs.
The original relation has no range.
Switching variables always removes the domain.
The inverse has no ordered pairs.

Show answer and explanation

Some inverse inputs have two possible outputs.

The original inputs

2

and

-2

both produce

4

. The inverse assigns input

4

both outputs

2

and

-2

.

Key terms

Relation: A set of pairings between inputs and outputs.
Function: A relation that gives exactly one output for each input.
Domain: The set of allowed inputs.
Range: The set of outputs produced by a relation or function.
One-to-one: A function in which different inputs always produce different outputs.
Inverse relation: The relation formed by reversing every ordered pair in the original relation.
Restricted domain: A smaller set of allowed inputs chosen to keep only part of a function's graph.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation RF6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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RF6 · Find and interpret inverse relations and restricted inverse functions

Alberta Grade 12 Mathematics

Reverse input and output, and restrict a domain when needed

What you will learn

1. Review: inputs, outputs, and pairings

2. Inverse relations and inverse functions

3. Finding, graphing, and interpreting inverses

4. A useful routine

A relation and its inverse

1. Invert a linear function

2. Restrict a quadratic to make an inverse function

3. Find an inverse relation that is not a function

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-1

About this lesson and its review