Inverse functions

In the parent chapter on composite and inverse functions, you have already seen the general ideas of a function, compositions like $f(g(x))$, and the basic meaning of “undoing” a function. Here we concentrate specifically on inverse functions: what they are, when they exist, how to find them, and how to work with them.

The idea of an inverse function

An inverse function reverses the action of a given function.

Suppose $f$ takes an input $x$ and produces an output $y$:
$$
y = f(x).
$$
An inverse function, usually written $f^{-1}$, takes that output $y$ and returns the original input:
$$
x = f^{-1}(y).
$$

Using $x$ as the variable again, we write this relationship in function notation as:
$$
f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x,
$$
whenever both sides make sense (i.e., for inputs in the appropriate domains).

Key idea: applying $f$ and then $f^{-1}$ (or $f^{-1}$ and then $f$) brings you back to where you started.

When does an inverse function exist?

Not every function has an inverse function. For $f$ to have an inverse function $f^{-1}$, it must be:

• One-to-one (injective): different inputs give different outputs. Formally, if $f(a) = f(b)$ then $a = b$.
• Equivalently: $f$ never takes the same value twice.

On a graph, there is a useful test:

• Horizontal line test: A function has an inverse function (defined on its entire domain) if and only if no horizontal line intersects its graph more than once.

Many familiar functions fail this test on their natural domains:

• $f(x) = x^2$ is not one-to-one on all real numbers, because $f(2) = f(-2) = 4$.
• $f(x) = \sin x$ is not one-to-one on all real numbers, because $\sin x$ repeats values periodically.

However, we can sometimes restrict the domain of a function to make it one-to-one so that it will have an inverse on that restricted domain. For example:

• $f(x) = x^2$ restricted to $x \ge 0$ is one-to-one.
• $\sin x$ restricted to $[-\pi/2, \pi/2]$ is one-to-one.

On these restricted domains, we can define inverse functions $\sqrt{x}$ and $\arcsin x$, respectively.

Domain and range in inverses

If $f$ has an inverse $f^{-1}$, then the roles of domain and range swap:

• If $f : A \to B$ (domain $A$, range $B$),
• Then $f^{-1} : B \to A$ (domain $B$, range $A$).

More concretely:

• The domain of $f^{-1}$ is the range of $f$.
• The range of $f^{-1}$ is the domain of $f$.

This is often tested in problems where you are given:

• The domain and range of $f$, and asked to find those of $f^{-1}$.
• Or you are given a graph of $f$ and need to determine possible domains/ranges for $f^{-1}$.

Graphs of inverse functions

Graphically, the inverse function corresponds to reflecting points across the line $y = x$.

If $(a,b)$ is a point on the graph of $f$, then $(b,a)$ is a point on the graph of $f^{-1}$. So:

• The graph of $f^{-1}$ is the mirror image of the graph of $f$ across the line $y = x$.

Practical uses:

• Given the graph of $f$, you can sketch $f^{-1}$ by flipping points over the line $y = x$.
• Given the graph of $f^{-1}$, you can recover the graph of $f$ by the same reflection, since the roles are symmetric.

This reflection picture also explains the domain–range swapping: the horizontal extent (domain) and vertical extent (range) of a graph are interchanged by reflecting over $y = x$.

Algebraic method for finding an inverse

For functions given by a formula, there is a standard algebraic procedure to find $f^{-1}$ (when it exists):

1. Start with $y = f(x)$.
2. Swap $x$ and $y$ (this reflects the idea of swapping input and output).
3. Solve the new equation for $y$ in terms of $x$.
4. Rename this solved expression as $f^{-1}(x)$.

This assumes that you can solve for $y$ uniquely; if not, the function may not be one-to-one on its given domain, or you may need a domain restriction to choose a single branch.

Example: A linear function

Let $f(x) = 3x - 5$.

1. Write $y = 3x - 5$.
2. Swap $x$ and $y$: $x = 3y - 5$.
3. Solve for $y$:
   $$
   x = 3y - 5 \
   x + 5 = 3y \
   y = \frac{x + 5}{3}.
   $$
4. Rename: $f^{-1}(x) = \dfrac{x + 5}{3}$.

You can verify:

• $f(f^{-1}(x)) = 3\cdot\dfrac{x+5}{3} - 5 = x+5-5 = x$,
• $f^{-1}(f(x)) = \dfrac{3x-5 + 5}{3} = \dfrac{3x}{3} = x$.

Example: A quadratic with restricted domain

Let $f(x) = x^2$ restricted to $x \ge 0$.

1. $y = x^2$, $x \ge 0$.
2. Swap $x$ and $y$: $x = y^2$.
3. Solve for $y$:
   $$
   x = y^2 \quad \Rightarrow \quad y = \pm\sqrt{x}.
   $$
   But $y$ corresponds to the original $x$, which had $x \ge 0$, so $y \ge 0$. Therefore, we choose:
   $$
   y = \sqrt{x}.
   $$
4. Rename: $f^{-1}(x) = \sqrt{x}$ with domain $x \ge 0$.

Checking domains and ranges:

• $f$: domain $[0,\infty)$, range $[0,\infty)$.
• $f^{-1}$: domain $[0,\infty)$, range $[0,\infty)$.

Here the function and its inverse share the same domain and range.

When the algebra gives multiple solutions

As in the quadratic example, sometimes solving for $y$ yields more than one expression (for example, $\pm\sqrt{x}$). This happens when the original function was not one-to-one on its full domain.

To define an inverse function (a single-valued function), you must:

• Restrict the domain of the original function so that it becomes one-to-one.
• Pick the branch of the solution that corresponds to that domain restriction.

Without this step, you do not get a proper inverse function; you only get a multi-valued inverse relation.

Verifying inverses with composition

In the parent chapter, you learned about function composition. That idea is crucial for checking inverses.

To verify that $g$ is the inverse of $f$:

• Compute $f(g(x))$ and simplify.
• Compute $g(f(x))$ and simplify.
• If both simplify to $x$ (for appropriate $x$ in their domains), then $g = f^{-1}$.

If either composition does not simplify to $x$ (or only does so on a restricted set), you may need to:

• Adjust the proposed formula for the inverse.
• Restrict the domain of $f$ or $g$.
• Or conclude that $g$ is not the inverse of $f$.

This “two-way composition equals $x$” condition is the defining property of inverse functions and is safer to rely on than just “solving for $y$” without checking.

Inverses and function behavior

Inverse functions also interact with properties such as monotonicity (whether a function is always increasing or always decreasing on an interval).

On an interval where $f$ is:

• Strictly increasing (never flat, never decreases), it is one-to-one and has an inverse on that interval.
• Strictly decreasing, it is also one-to-one and has an inverse on that interval.

On such intervals, the inverse function is also strictly monotonic (increasing if $f$ is increasing, decreasing if $f$ is decreasing), but with the roles of input and output switched.

This connection between inverses and monotonicity is particularly important for:

• Choosing appropriate domains for inverse trigonometric functions.
• Understanding the shapes of graphs of inverse functions.

Inverse transformations and solving equations

Inverse functions are a powerful tool for solving equations of the form
$$
f(x) = c
$$
for $x$.

If $f$ has an inverse and $c$ is in the range of $f$, then
$$
x = f^{-1}(c)
$$
is the solution.

In practice:

• To solve $3x - 5 = 7$, you are applying the inverse of $x \mapsto 3x - 5$, which is $x \mapsto \dfrac{x+5}{3}$:
  $$
  x = \frac{7 + 5}{3} = 4.
  $$
• To solve $x^2 = 9$ with the restriction $x \ge 0$, you are applying the inverse of $x \mapsto x^2$ on $[0,\infty)$, which is $\sqrt{x}$:
  $$
  x = \sqrt{9} = 3.
  $$

This perspective becomes more important in later topics (for example, using logarithms as inverses of exponentials to solve exponential equations, or using inverse trigonometric functions to solve trigonometric equations on specified intervals).

Common pitfalls and clarifications

1. $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$.

The notation looks similar, but they mean different things:

• $f^{-1}(x)$: the inverse function evaluated at $x$.
• $\dfrac{1}{f(x)}$: the reciprocal of the value $f(x)$.

These are usually completely different functions. For example:

• If $f(x) = 3x - 5$, then $f^{-1}(x) = \dfrac{x+5}{3}$, but $\dfrac{1}{f(x)} = \dfrac{1}{3x-5}$.

2. A function can have at most one inverse function.

If an inverse function exists, it is unique. You cannot have two different functions that both act as inverses on the same domain and range.

However, you can have multiple inverse relations (for example, $y = \pm\sqrt{x}$ for $x^2$), corresponding to different domain restrictions of the original function.

3. Domain issues matter.

When checking whether $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$, you must pay attention to where these are valid:

• $f^{-1}(f(x)) = x$ holds for $x$ in the domain of $f$.
• $f(f^{-1}(x)) = x$ holds for $x$ in the domain of $f^{-1}$ (the range of $f$).

Mistakes often come from ignoring these domain restrictions.

4. Some functions are their own inverses.

A function $f$ is its own inverse if $f(f(x)) = x$ for all $x$ in the domain. In that case, $f^{-1} = f$.

On a graph, such functions are symmetric with respect to the line $y = x$ and also coincide with their reflection.

Example: $f(x) = \dfrac{1}{x}$ for $x \ne 0$ (on certain domains) and $f(x) = x$ itself.

Summary of key points

• An inverse function $f^{-1}$ reverses $f$: $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$ (on appropriate domains).
• A function has an inverse function exactly when it is one-to-one (passes the horizontal line test); sometimes this requires restricting the domain.
• The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.
• The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.
• To find an inverse algebraically: write $y = f(x)$, swap $x$ and $y$, solve for $y$, and rename as $f^{-1}(x)$, checking domain issues.
• Composition is the definitive way to verify inverses: both $f(f^{-1}(x))$ and $f^{-1}(f(x))$ must simplify to $x$ where defined.
• $f^{-1}(x)$ is not the same as $1/f(x)$; they are different concepts.

Understanding inverse functions at this level prepares you for important later topics, such as logarithms as inverses of exponentials, inverse trigonometric functions, and transformations between different coordinate systems.