Inverse functions and restricted domains | Year 12

01 / LEARN

The idea, step by step

Undo input and output

An inverse reverses the original mapping, so its domain is the original range and its range is the original domain. Write y=f(x), solve for x, then swap variable names. The graph is reflected in y=x.

The notation f inverse is not the reciprocal 1/f. Check the proposed inverse by composing in both orders on their appropriate domains.

Select the right branch

Solving a square relation introduces two possible roots. The original domain chooses one branch; using both would not produce an inverse function.

For logarithmic inverses, a positive logarithm argument corresponds to the original exponential range. Preserve strict versus inclusive endpoints when swapping domains and ranges.

A common trap

Writing both square-root branches as one inverse function.

Instead: Use the branch selected by the original restricted domain.

02 / WORKED EXAMPLES

See the thinking

Example 1

The increasing parabola branch

Find the inverse of $f (x) = {(x - 3)}^{2} + 2$ , restricted to $x \geq 3$ . Give its domain and range.

$y - 2 = {(x - 3)}^{2}$ .
The chosen branch has x-3 nonnegative, so take the positive square root.
Swap the original domain and range.

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

; domain

[2, \infty)

, range

[3, \infty)

Example 2

A self-inverse rational rule

Find the inverse of $f (x) = x / (x - 1)$ , $x \neq 1$ .

$y (x - 1) = x$ gives $x (y - 1) = y$ .
The function is its own inverse on the stated domain.

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

x \neq 1

Check your understanding

If f has domain $[3, \infty)$ and range $[2, \infty)$ , what is the domain of its inverse?

Show answer

$[2, \infty)$ .

Input and output sets exchange roles.

03 / WORKSHEET

Your turn

6 questions · 20 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the inverse of $f (x) = (2 x + 1) / 3$ .
3 marks
Find the inverse of $f (x) = {(x - 3)}^{2} + 2$ , restricted to $x \geq 3$ . Give its domain and range.
4 marks
Find the inverse of $f (x) = {(x - 3)}^{2} + 2$ when $x \leq 3$ .
3 marks
Find the inverse of $f (x) = x / (x - 1)$ , $x \neq 1$ .
4 marks
Find the inverse of $f (x) = 3 e^{x} - 4$ and its domain.
3 marks
Find a domain restriction for $f (x) = ∣ x + 1 ∣$ that gives inverse $x - 1$ , and state the inverse domain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the inverse of $f (x) = (2 x + 1) / 3$ .
3 marks
Write $y = (2 x + 1) / 3$ .
Solve for x: $x = (3 y - 1) / 2$ , then swap the variable names.
$f^{- 1} (x) = (3 x - 1) / 2$ .
Find the inverse of $f (x) = {(x - 3)}^{2} + 2$ , restricted to $x \geq 3$ . Give its domain and range.
4 marks
$y - 2 = {(x - 3)}^{2}$ .
The chosen branch has x-3 nonnegative, so take the positive square root.
Swap the original domain and range.
$f^{- 1} (x) = 3 + \sqrt{x - 2}$ ; domain $[2, \infty)$ , range $[3, \infty)$ .
Find the inverse of $f (x) = {(x - 3)}^{2} + 2$ when $x \leq 3$ .
3 marks
The original branch has x-3 nonpositive.
Choose the negative square-root branch before swapping variables.
$f^{- 1} (x) = 3 - \sqrt{x - 2}$ , $x \geq 2$ .
Find the inverse of $f (x) = x / (x - 1)$ , $x \neq 1$ .
4 marks
$y (x - 1) = x$ gives $x (y - 1) = y$ .
The function is its own inverse on the stated domain.
$f^{- 1} (x) = x / (x - 1)$ , $x \neq 1$ .
Find the inverse of $f (x) = 3 e^{x} - 4$ and its domain.
3 marks
Solve $y + 4 = 3 e^{x}$ for x.
The original range is greater than -4.
$ln ((x + 4) / 3)$ , $x > - 4$ .
Find a domain restriction for $f (x) = ∣ x + 1 ∣$ that gives inverse $x - 1$ , and state the inverse domain.
3 marks
On the right branch, f(x)=x+1.
The branch range is nonnegative.
Restrict f to $x \geq - 1$ ; then $f^{- 1} (x) = x - 1$ for $x \geq 0$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist functions and sketching graphs. Calculus-based analysis assumes concurrent Methods differentiation; real-polynomial division detail is taught in the separate Real Polynomials chapter.

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