Inverse Functions (Grade 12)

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 = x3 + 2.
Make y the subject.
4
h⁻¹(x) = 13x + 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 = x2.
3
y = ±x2.
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) = x2; if x ≤ 0 then f⁻¹(x) = −x2.

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

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).

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 = …
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x = 3y − 6 → y = 13x + 2
2
Determine the inverse of f(x) = 2x².
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x = 2y² → y = ±x2
3
Is the inverse of f(x) = 2x² a function? Give a reason.
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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.
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x ≥ 0 (or x ≤ 0)
5
Determine the equation of k⁻¹ if k(x) = 2^x.
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y = log₂ x
6
Write down the domain and range of k⁻¹ if k(x) = 2^x.
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Domain x > 0; range y ∈ ℝ
7
The graph of f⁻¹ is the reflection of f in which line?
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y = x
8
The point (2 ; 7) lies on f. Write down a point on f⁻¹.
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(7 ; 2)