Connect inverse composition with the identity function

Ontario Grade 12 Mathematics

Characteristics of Functions

How applying a function and its inverse brings an input back to where it started

A function can change an input according to a rule. Its inverse reverses that change. In this lesson, you will connect that reversal to the identity function, which leaves every allowed input unchanged. The main idea is that composing a function with its inverse returns the starting value, as long as the input belongs to the appropriate domain.

What you will learn

Explain how an inverse function undoes the action of a function.
Connect composition of a function and its inverse to the identity function.
Verify the connection using numerical values and symbolic notation.
Recognize that each composition must use inputs allowed by the functions involved.

1. Prerequisite bridge: functions, inverses, and composition

A function assigns exactly one output to each input in its domain. The domain is the set of inputs that are allowed. The range is the set of outputs the function produces. For example, if a function doubles an input and adds three, then an input of four produces eleven.

An inverse function reverses the input-output pairs of the original function. If the original function sends four to eleven, its inverse sends eleven back to four. The inverse is written with a superscript

^{-1}

: if the function is

f

, its inverse is

f^{-1}

. This notation does not mean that the function is raised to the power of negative one.

Composition means using one function's output as the next function's input. The notation

(f\circ g)(x)

means first apply

g

to

x

, then apply

f

to the result. The order matters:

(f\circ g)(x)

and

(g\circ f)(x)

can give different results.

(f\circ g)(x)=f(g(x))

An inverse reverses the input-output pairs of a function.
In a composition, the function nearest the input is applied first.
The output from the first function must be an allowed input for the next one.

2. Plain-language connection: undoing a function

The identity function leaves its input unchanged. For example, it sends five to five and sends any other allowed value back to itself. Its rule is often written as

I(x)=x

.

Now consider a function and its inverse. If you apply the function first, then apply its inverse, the inverse undoes the change. The final output is the original input. This composition acts like the identity function.

The order can also be reversed. Applying the inverse first and then the original function also returns the starting input, when that input is in the domain of the inverse. These statements describe two compositions, so it is important to check the order and the allowed inputs rather than assuming the expressions are interchangeable.

(f^{-1}\circ f)(x)=x,\qquad (f\circ f^{-1})(x)=x

The identity function returns its input unchanged.
Function followed by its inverse returns an input to itself.
Inverse followed by function also returns an input to itself on the inverse's domain.

3. Representing the connection with values and symbols

A small input-output list makes the reversal visible. Suppose a function pairs

2

with

7

,

3

with

9

, and

4

with

11

. Its inverse swaps each input and output: it pairs

7

with

2

,

9

with

3

, and

11

with

4

. Starting at

3

, the original function gives

9

, and the inverse takes

9

back to

3

. That is the identity action on this input.

In symbols, the same idea is expressed through composition. The expression

f^{-1}(f(x))

means that

f

acts first and

f^{-1}

acts second. The result is

x

. In the other order,

f(f^{-1}(x))

also returns

x

, provided the input is allowed for that composition.

The domain detail matters. In

f^{-1}(f(x))

, the starting value must be in the domain of

f

. In

f(f^{-1}(x))

, the starting value must be in the domain of

f^{-1}

, which corresponds to the range of

f

. The compositions return the input on those appropriate sets.

f^{-1}(f(x))=x

The inverse swaps inputs and outputs in an input-output list.
The order of functions in the written composition shows which acts first.
The identity result applies to inputs allowed in the composition.

4. Guided example and application

Use a simple linear function to check both compositions. A linear function has the form of a straight-line rule. Here, each input is doubled and then three is added. To find its inverse, reverse those operations: subtract three, then divide by two.

This example shows more than a calculation. It connects a function, its inverse, and the identity function in both orders. You can test the connection with a value and then express the result for a general input.

To undo a sequence of operations, reverse the order and undo each operation.
Check both composition orders when connecting an inverse to the identity.

A function and its inverse reverse the pairs

Input to $f$	Output from $f$	Input to $f^{-1}$	Output from $f^{-1}$
2	7	7	2
3	9	9	3
4	11	11	4

Worked example

Verify both inverse compositions

Let

f(x)=2x+3

. Find

f^{-1}(x)

, then show how composing the functions gives the identity on the appropriate inputs.

Reverse the operations
The function doubles its input and adds three. To undo it, subtract three from the output and divide by two. Therefore, the inverse rule is:
$f^{-1}(x)=\frac{x-3}{2}$
Compose the inverse after the function
In $f^{-1}(f(x))$ , apply $f$ first. Substitute $2x+3$ into the inverse rule. The subtraction cancels the added three, and division by two cancels the doubling.
$f^{-1}(f(x))=\frac{(2x+3)-3}{2}=x$
Compose the function after the inverse
In $f(f^{-1}(x))$ , apply the inverse first. Substitute $(x-3)/2$ into the rule for $f$ . Multiplying by two and adding three returns the original input.
$f(f^{-1}(x))=2\left(\frac{x-3}{2}\right)+3=x$
Check with a value
For example, $f(4)=11$ , and the inverse sends $11$ back to $4$ . This numerical check matches the symbolic identity result.
$f^{-1}(f(4))=f^{-1}(11)=4$

Answer: The inverse is

f^{-1}(x)=\frac{x-3}{2}

. Both compositions return the input, so each acts as the identity on its appropriate domain.

Check: The two rules undo one another: doubling and adding three is reversed by subtracting three and dividing by two.

Common mistakes and how to avoid them

Treating

f^{-1}(x)

as the reciprocal of

f(x)

.

Correction: The notation

f^{-1}

names the inverse function. It reverses the input-output pairs; it does not mean

1/f(x)

.

Reversing the order of a composition.

Correction: Read from the inside out. In

f^{-1}(f(x))

, apply

f

first and then its inverse.

Saying the composition returns every possible number without checking the input.

Correction: The starting input must be in the domain of the first function used. For

f(f^{-1}(x))

, it must also be an output that

f

can produce.

Assuming that a composition equal to

x

means the two functions are identical.

Correction: The composition uses both functions in sequence. The result is the identity action because the inverse undoes the original function.

Lesson summary

The identity function leaves each allowed input unchanged.
An inverse reverses the input-output pairs of the original function.
Composing a function with its inverse returns the starting input.
The two composition orders are written separately, and each has appropriate input restrictions.

Check your understanding

Question 1

If

g(5)=12

, what must

g^{-1}(12)

equal?

5
12
17
The value cannot be determined.

Show answer and explanation

5

The inverse reverses the pair. Since

g

sends

5

to

12

,

g^{-1}

sends

12

back to

5

.

Question 2

Which expression means apply

h

first and then apply its inverse?

$(h^{-1}\circ h)(x)$
$(h\circ h^{-1})(x)$
$h(x)+h^{-1}(x)$
$h^{-1}(h^{-1}(x))$

Show answer and explanation

(h^{-1}\circ h)(x)

The inside function is applied first. In

(h^{-1}\circ h)(x)

,

h

acts before

h^{-1}

.

Question 3

Suppose

p^{-1}(p(x))=x

for an allowed input

x

. What does this result show?

The composition acts as the identity on that input.
The two functions have the same rule.
The output must be zero.
The inverse changes the input a second time.

Show answer and explanation

The composition acts as the identity on that input.

Returning

x

unchanged is exactly the action of the identity function.

Key terms

Composition: A process that applies one function and then uses its output as the input to another function.
Domain: The set of inputs for which a function is defined.
Identity function: A function that returns each allowed input unchanged.
Inverse function: A function that reverses the input-output pairs of another function.
Range: The set of outputs produced by a function.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D2.7. 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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