Inverse Functions — Grade 12 Maths (CAPS)

Finding the inverse of a line, a parabola and an exponential function, why y = x² needs a restricted domain, and how to state whether an inverse is a function.

The inverse undoes the function. Every (x ; y) on f becomes (y ; x) on f⁻¹, so the graph of f⁻¹ is the reflection of f in the line y = x.

1The method (three steps, every time)

Write the function as y = …
Swap x and y.
Make y the subject — that is f⁻¹.

h(x) = 3x − 6 and its inverse. Each is the mirror image of the other in the dashed line y = x.

2Inverse of a straight line

Worked Exampleh(x) = 3x − 6. Determine h⁻¹

1
y = 3x − 6.
2
Swap: x = 3y − 6.
3
3y = x + 6 → y = x/3 + 2.
Make y the subject.
4
h⁻¹(x) = 1/3x + 2.
The inverse of a line is always a line.

y = 2x² restricted to x ≥ 0, with its inverse. The point (2 ; 8) becomes (8 ; 2).

3Inverse of a parabola

Worked Examplef(x) = 2x². Determine f⁻¹

1
y = 2x² → swap: x = 2y².
2
y² = x/2.
3
y = ±√(x/2).
Square roots give TWO answers — the ± is essential.

⚠️

Because of the ±, the inverse of an unrestricted parabola is not a function — one x-value gives two y-values, so it fails the vertical line test.

To force it to be a function you restrict the domain of f. If x ≥ 0 then f⁻¹(x) = √(x/2); if x ≤ 0 then f⁻¹(x) = −√(x/2).

y = 2^x and y = log₂ x are reflections of each other in y = x. The y-intercept (0 ; 1) becomes the x-intercept (1 ; 0).

4Inverse of an exponential

For b > 0 and b ≠ 1:
f(x) = b^x ⇔ f⁻¹(x) = log_b x

Worked Examplek(x) = 2^x. Determine k⁻¹

1
y = 2^x → swap: x = 2^y.
2
y = log₂ x.
A logarithm is exactly the tool for making an exponent the subject.

5Domain and range swap over

domain of f⁻¹ = range of f
range of f⁻¹ = domain of f

Worked ExampleState the domain and range of k⁻¹ if k(x) = 2^x

1
k has domain x ∈ ℝ and range y > 0.
An exponential is never zero or negative.
2
So k⁻¹ has domain x > 0 and range y ∈ ℝ.
They simply trade places.

💡

Quick check: the y-intercept of f becomes the x-intercept of f⁻¹. For k(x) = 2^x the point (0 ; 1) becomes (1 ; 0).

6The definition, and why it decides everything here

A function is a rule in which each input value has exactly one output value — the output uniquely depends on the input. You met that in Grade 10; it matters again now because it is the whole reason an inverse can fail.

Reflecting a graph in the line y = x swaps the inputs and the outputs. So a function whose own graph passes the vertical line test may produce a reflection that does not — and that reflection is then a relation, not a function.

One-to-one (every output used once, like y = ax + q): the inverse IS a function.
Many-to-one (two inputs share an output, like y = ax²): the inverse is one-to-many, so it is NOT a function.
That is exactly why the domain of y = ax² must be restricted to x ≥ 0 or x ≤ 0 before its inverse can be called a function.

💡

The test transfers neatly: a graph is a function if no vertical line cuts it twice; its inverse is a function if no horizontal line cuts the original twice.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Determine the equation of h⁻¹, the inverse of h(x) = 3x − 6, in the form y = …
Show answer ▾Hide answer ▴
x = 3y − 6 → y = 1/3x + 2
2
Determine the inverse of f(x) = 2x².
Show answer ▾Hide answer ▴
x = 2y² → y = ±√(x/2)
3
Is the inverse of f(x) = 2x² a function? Give a reason.
Show answer ▾Hide answer ▴
No — because of the ±, one x-value gives two y-values, so it fails the vertical line test.
4
Restrict the domain of f(x) = 2x² so that f⁻¹ is a function.
Show answer ▾Hide answer ▴
x ≥ 0 (or x ≤ 0)
5
Determine the equation of k⁻¹ if k(x) = 2^x.
Show answer ▾Hide answer ▴
y = log₂ x
6
Write down the domain and range of k⁻¹ if k(x) = 2^x.
Show answer ▾Hide answer ▴
Domain x > 0; range y ∈ ℝ
7
The graph of f⁻¹ is the reflection of f in which line?
Show answer ▾Hide answer ▴
y = x
8
The point (2 ; 7) lies on f. Write down a point on f⁻¹.
Show answer ▾Hide answer ▴
(7 ; 2)
9
State the definition of a function.
Show answer ▾Hide answer ▴
A rule in which every input value has exactly ONE output value ✓✓
10
Explain, using the definition, why the inverse of f(x) = 3x² is not a function unless the domain is restricted.
Show answer ▾Hide answer ▴
f is many-to-one — f(−2) and f(2) both give 12 ✓; reflecting swaps inputs and outputs, so one input would map to two outputs ✓; restricting to x ≥ 0 makes f one-to-one, so the inverse is a function ✓
11
Which line test tells you whether the INVERSE of a graph will be a function?
Show answer ▾Hide answer ▴
The horizontal line test applied to the ORIGINAL graph ✓; if a horizontal line cuts it more than once, the inverse is not a function ✓