A Shortcut for the Inverse

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A sequence of reversible operations can be undone in reverse order. This is a useful shortcut when the function is built from simple one-to-one steps and their domains are respected.

Visual guide

VISUAL GUIDEUndoing a linear rule

f(x) = 2x + 1
f⁻¹(x) = (x − 1)/2
Reflection line y = x

For f(x) = 2x + 1, the inverse is (x − 1)/2. Points (0, 1) and (1, 0) exchange coordinates across y = x. The inverse reverses the input/output relationship; it is not the reciprocal 1/f(x).

Method

If $f=g\circ h$ and both maps are bijective between their specified sets, then

f^{-1}=h^{-1}\circ g^{-1}.

The last operation performed is the first operation undone.

Worked example

For $f(x)=2(x-3)^3+5$ , first subtract $3$ , cube, multiply by $2$ , then add $5$ . Reverse these operations: subtract $5$ , divide by $2$ , take a cube root, add $3$ .

f^{-1}(x)=3+\sqrt[3]{(x-5)/2}.

Cube roots are single-valued over the reals, so no branch restriction is needed.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

For f(x)=ax+b with a≠0, what is f⁻¹(x)?

ax−b

(x+b)/a

(x−b)/a

Hint 1 · Find a starting point

An affine rule is undone in reverse order.

Hint 2 · Take the next step

Subtract b and then divide by a.

Show the reasoning

Answer: (x−b)/a

f((x−b)/a)=x, which checks the inverse formula.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

Undoing in the same order is wrong. Squaring also requires a domain restriction before a square root can undo it.

Check your understanding

Invert $f(x)=5x+7$ .

Show answer

First subtract $7$ , then divide by $5$ : $(x-7)/5$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →