Inverse functions and their derivatives

One-to-one functions and inverse functions, reflection in y = x, swapping domain and range, restricting a domain to create an inverse, and the derivative of an inverse function using dy/dx = 1 ÷ dx/dy.

Inverse functions

A function has an inverse function exactly when it is one-to-one (it passes the horizontal line test). The inverse $f^{-1}$ undoes $f$: $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$. Its graph is the reflection of $y = f(x)$ in the line $y = x$, so the domain and range swap.

A function and its inverse

y = x³/8 + 1 (solid) and its inverse y = 2∛(x − 1) (dashed) are reflections of each other in y = x.

Try this

The solid curve passes through $(2, 2)$. Why does the dashed one pass through it too?

Example 1: restricting the domain●●

Find an inverse function for $f(x) = (x - 2)^2$, $x \ge 2$.

On $x \ge 2$ the function is increasing, so it is one-to-one.

Swap $x$ and $y$: $x = (y - 2)^2$, so $y = 2 + \sqrt x$ (taking the positive root, since $y \ge 2$). So $f^{-1}(x) = 2 + \sqrt x$, $x \ge 0$.

The derivative of an inverse function

Derivative of an inverse

$$\frac{dy}{dx} = \frac{1}{\;\dfrac{dx}{dy}\;}$$

Reflecting in $y = x$ swaps rise and run, so gradients at corresponding points are reciprocals.

Example 2●●●

$f(x) = x^3 + x$. Find the gradient of $y = f^{-1}(x)$ at the point where $x = 2$.

$f(1) = 2$, so the point $(2, 1)$ lies on $y = f^{-1}(x)$.

$f'(x) = 3x^2 + 1$, so $f'(1) = 4$.

The gradient of the inverse at $(2, 1)$ is $\dfrac14$.

Flashcards

When does $f$ have an inverse function?

When it is one-to-one.

Graph of $f^{-1}$

The reflection of $y = f(x)$ in $y = x$.

Gradient of the inverse

$\dfrac{dy}{dx} = 1 \Big/ \dfrac{dx}{dy}$

Practice

$f(x) = x^3 + x$. Find the gradient of $y = f^{-1}(x)$ at $x = 2$.
For the same $f$, find the gradient of $y = f^{-1}(x)$ at $x = 10$.
$f^{-1}(x) = 2 + \sqrt x$. Find $f^{-1}(9)$.
$y = e^{2x}$, so $x = \tfrac12\ln y$. Find $\dfrac{dx}{dy}$ when $y = 1$.
$f(4) = 7$ and $f'(4) = \tfrac13$. Find the gradient of $y = f^{-1}(x)$ at $x = 7$.
Which function has an inverse function?
The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in:
The domain of $f^{-1}$ is:
$f$ has a stationary point at $(1, 3)$. At $(3, 1)$, the graph of $f^{-1}$ has: