Inverse relations and functions

Form the inverse relation by swapping x and y, graph it by reflecting in y = x, swap the domain and range, handle restrictions, and use the horizontal line test.

The inverse relation

An inverse undoes a process: the inverse of "multiply by $7$" is "divide by $7$". For a relation, the inverse simply swaps the roles of $x$ and $y$.

Example 1●

Find the inverse of $y = 2x + 4$.

$x = 2y + 4$

Swap x and y.

$y = \dfrac{x - 4}{2}$

Make y the subject.

A cubic and its inverse relation

Orange: y = x³ − kx. Blue: the inverse relation, its reflection in the dashed line y = x.

Try this

Find the value of $k$ where the cubic stops having turning points. What happens to the blue graph there?

Is the inverse a function?

The inverse is a function when it passes the vertical line test. Vertical lines on the inverse come from horizontal lines on the original, so:

Example 2●●

Find the inverse of $y = x^2$. Is it a function?

$x = y^2$, so $y = \sqrt{x}$ or $y = -\sqrt{x}$.

Not a function: $y = x^2$ fails the horizontal line test.

The line y = 4 meets y = x² twice, at x = ±2.

Example 3: a cube root●

Find the inverse of $y = x^3$.

$x = y^3$, so $y = \sqrt[3]{x}$. Every horizontal line meets $y = x^3$ once, so the inverse is a function.

Restrictions

A restriction on $x$ in the original becomes a restriction on $y$ in the inverse, and that decides which square root to take.

Example 4●●●

Find the inverse of $y = x^2 - 1$ with $x \le 0$, and state its domain and range.

$x = y^2 - 1$ with $y \le 0$

Swap x and y, including the restriction.

$y = -\sqrt{x + 1}$

y ≤ 0, so take the negative square root.

Domain $x \ge -1$, range $y \le 0$.

The original has domain x ≤ 0 and range y ≥ −1. Swap them.

Flashcards

Inverse relation

Swap $x$ and $y$. The graph reflects in $y = x$.

Domain of the inverse

The range of the original.

Horizontal line test

The inverse is a function exactly when every horizontal line meets the original at most once.

Practice

The graph of an inverse relation is the reflection of the original in:
The domain of the inverse relation is:
The inverse of $y = 2x + 4$ is:
$(3, 7)$ lies on a relation. Which point lies on the inverse? Enter its $x$-coordinate.
The inverse relation of $y = x^2$ is:
The inverse relation is a function exactly when the original:
A function has domain $x \ge 2$ and range $y \ge -1$. The domain of its inverse is $x \ge c$. Find $c$.
The inverse of $y = x^3$ is:
The inverse of $y = x^2 - 1$, $x \le 0$, is:
The inverse of $y = 2x + 4$ is evaluated at $x = 10$. Find the result.