Inverse functions: reflecting in y = x

What is it about?

The inverse function $f^{-1}$ reverses $f$: from $f(3) = 5$ we get $f^{-1}(5) = 3$. This only works if every value occurs exactly once – every horizontal line meets the graph at most once. That is certainly the case if $f$ is strictly monotonic. $x^2$ on all of $\mathbb{R}$ is not invertible ($f(-2) = f(2) = 4$); restricted to $x \geq 0$ its inverse is $\sqrt{x}$, restricted to $x \leq 0$ it is $-\sqrt{x}$. Monotonicity is sufficient but not necessary: $\frac{1}{x}$ is invertible (it is even its own inverse).

To find $f^{-1}$: solve $y = f(x)$ for $x$, then swap $x$ and $y$. Graphically this is a reflection in $y = x$: $P(a \mid b)$ becomes $P'(b \mid a)$, and domain and range swap. Writing $D_f$ for the domain and $R_f$ for the range of $f$: $D_{f^{-1}} = R_f$, $R_{f^{-1}} = D_f$ (German textbooks write $W_f$ for the range, from Wertemenge). This is how $\ln x$ arises from $e^x$: $D_f = \mathbb{R}$ and $R_f = \mathbb{R}^+$, so $\ln x$ is defined only for $x > 0$.

Reflecting the tangent swaps the legs of the slope triangle, so for $b = f(a)$ and $f'(a) \neq 0$:

If $f'(a) = 0$, the inverse has a vertical tangent at $(b \mid a)$, e.g. $0.5x^3$ at $0$.

Try it

1. Square root: drag $P$ or $P'$, then press “Reflect in y = x” and watch the graph flip over, the intervals on the axes and the columns of the table.
2. Line: press “Working step by step”.
3. x² on all of ℝ: sweep the horizontal line, then switch to a branch.
4. eˣ and ln x and the vertical tangent.
5. 1/x and, as an outlook, restricted sine.

Tasks

Find the inverse of $f(x) = 2x - 1$ and $f^{-1}(5)$.

For $f(x) = x^2$, $x \geq 0$, and $P(1.5 \mid 2.25)$: find the slope of $\sqrt{x}$ at $2.25$.

Why is $\sqrt[3]{2x}$ not differentiable at $0$? Find $\left(f^{-1}\right)'(4)$ for $f(x) = 0.5x^3$.

Ideas for the classroom

• Use the views in order: “Invertible?” → “Reflect in y = x” → “Derivative of f⁻¹” leading to $(\ln x)' = \frac{1}{x}$.
• Draw a graph on a transparency and flip it along $y = x$, then compare with the animation.
• With show=0 and inv=0 the pictures work as exercises.
• Common stumbling blocks:
  • confusing $f^{-1}(x)$ with $\frac{1}{f(x)}$;
  • forgetting to swap $x$ and $y$, or forgetting the restricted domain (sign of $-\sqrt{x}$);
  • thinking “strictly monotonic” requires $f' > 0$ everywhere ($0.5x^3$ has $f'(0) = 0$);
  • evaluating $f'$ at $b$ instead of $a$ in $\left(f^{-1}\right)'(b) = \frac{1}{f'(a)}$.