3.7 Inverse Functions

Inverse Functions

Inverse functions undo each other's operations. If $f(x)$ takes an input and produces an output, then its inverse $f^{-1}(x)$ takes that output and returns the original input. This concept is central to solving equations, understanding function composition, and working with more advanced algebra topics.

Inverse Function Determination

Two functions $f(x)$ and $g(x)$ are inverses if composing them in either order gives back the original input $x$. Both conditions must hold:

• $f(g(x)) = x$ for all $x$ in the domain of $g$
• $g(f(x)) = x$ for all $x$ in the domain of $f$

To check whether two functions are inverses, compute both compositions and simplify. If both reduce to $x$, the functions are inverses.

Example: Let $f(x) = 2x + 6$ and $g(x) = \frac{x - 6}{2}$.

• $f(g(x)) = 2\left(\frac{x - 6}{2}\right) + 6 = (x - 6) + 6 = x$ ✓
• $g(f(x)) = \frac{(2x + 6) - 6}{2} = \frac{2x}{2} = x$ ✓

Both compositions equal $x$, so $f$ and $g$ are inverses.

Domain and Range of Inverses

Because inverse functions swap inputs and outputs, their domains and ranges also swap:

• The domain of $f(x)$ becomes the range of $f^{-1}(x)$
• The range of $f(x)$ becomes the domain of $f^{-1}(x)$

For example, if $f(x) = x^2$ is restricted to $[0, \infty)$, its range is $[0, \infty)$. The inverse $f^{-1}(x) = \sqrt{x}$ then has domain $[0, \infty)$ and range $[0, \infty)$.

One-to-one requirement: A function must be one-to-one (each output comes from exactly one input) to have an inverse. You can test this with the horizontal line test: if any horizontal line crosses the graph more than once, the function is not one-to-one and doesn't have an inverse without restricting its domain.

This is exactly why $f(x) = x^2$ on all of $\mathbb{R}$ has no inverse: both $x = 3$ and $x = -3$ produce $y = 9$, so there's no unique way to reverse the operation. Restricting the domain to $[0, \infty)$ fixes this problem.

Calculation of Inverse Functions

Finding the inverse of a function follows a consistent set of steps:

1. Replace $f(x)$ with $y$
2. Swap $x$ and $y$
3. Solve the new equation for $y$
4. Write the result as $f^{-1}(x)$

Example: Find the inverse of $f(x) = 3x - 4$.

1. $y = 3x - 4$

2. $x = 3y - 4$

3. $x + 4 = 3y$, so $y = \frac{x + 4}{3}$

4. $f^{-1}(x) = \frac{x + 4}{3}$

You can verify by composing: $f(f^{-1}(x)) = 3\left(\frac{x+4}{3}\right) - 4 = x + 4 - 4 = x$. It checks out.

Graphing Inverses from Originals

The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y = x$. This makes sense visually because reflecting across $y = x$ swaps every point's coordinates.

Reflection process:

1. Pick several points on the graph of $f(x)$. For example, if $f(x) = x^2$ (with $x \geq 0$), you might choose $(0, 0)$, $(1, 1)$, and $(2, 4)$.
2. Swap the $x$ and $y$ coordinates of each point: $(0, 0)$, $(1, 1)$, $(4, 2)$.
3. Plot these new points and connect them to sketch the inverse. Here, you'd get the graph of $f^{-1}(x) = \sqrt{x}$.

The key property at work: if $(a, b)$ is on the graph of $f(x)$, then $(b, a)$ is on the graph of $f^{-1}(x)$. The line $y = x$ acts as the mirror between the two graphs, and the function and its inverse will always be symmetric about that line.