Functions • Topic 4 of 4

Inverse Functions

The inverse f-inverse undoes f: if f sends a to b, then f-inverse sends b back to a, so f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Only a bijection (one-one and onto) has an inverse, which is why CAT often pairs "find the inverse" with a check on those properties. The mechanical recipe is reliable: write y = f(x), swap x and y, then solve for y — that solved expression is f-inverse. For a linear f(x) = ax + b the inverse is (x − b)/a; for f(x) = 1/x the inverse is itself. Graphically, f and f-inverse are mirror images across the line y = x, so their graphs reflect into each other. Two facts win CAT questions fast: an involution satisfies f(f(x)) = x, meaning f is its own inverse (e.g. f(x) = 1/x, f(x) = a − x, f(x) = (ax + b)/(cx − a)); and the inverse of a composite reverses the order, (f∘g)⁻¹ = g⁻¹∘f⁻¹. When the domain is restricted, remember the inverse swaps the roles of domain and range.

✅ Solved examples

1. Find the inverse of f(x) = 3x − 7.
y = 3x − 7 ⇒ x = (y + 7)/3. Swap names: f⁻¹(x) = (x + 7)/3.
2. Find the inverse of f(x) = (2x + 3)/(x − 1).
y(x − 1) = 2x + 3 ⇒ x(y − 2) = y + 3 ⇒ x = (y + 3)/(y − 2). So f⁻¹(x) = (x + 3)/(x − 2).
3. Show f(x) = 5 − x is its own inverse.
f(f(x)) = 5 − (5 − x) = x. Since f(f(x)) = x, f equals f⁻¹ (an involution).
4. If f(x) = 2x + 1 and g(x) = x³, find (f∘g)⁻¹(x).
f(g(x)) = 2x³ + 1. Set y = 2x³ + 1 ⇒ x³ = (y − 1)/2 ⇒ x = ∛((y − 1)/2). So (f∘g)⁻¹(x) = ∛((x − 1)/2).

✏️ Practice — try these, take hints as needed

1. Find f⁻¹(x) for f(x) = 4x + 9.
Write y = 4x + 9.
Solve x = (y − 9)/4.
Swap x and y.
(x − 9)/4
2. Find f⁻¹(x) for f(x) = (x − 2)/(x + 2).
Cross-multiply y(x + 2) = x − 2.
Group x terms: x(y − 1) = −2 − 2y.
Solve for x.
(2 + 2x)/(1 − x)
3. Is f(x) = 1/x its own inverse?
Compute f(f(x)).
f(1/x) = 1/(1/x).
Simplify.
Yes, f(f(x)) = x
4. If f⁻¹(x) = (x + 5)/2, find f(x).
The inverse of the inverse is f.
Set y = (x + 5)/2 and solve for x.
x = 2y − 5, then swap.
2x − 5
5. f(x) = x² on domain [0, ∞). Find f⁻¹(x).
On [0, ∞) the function is one-one.
y = x² ⇒ x = √y (non-negative root).
Swap names.
√x

📝 Topic test — 8 questions

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