5.7 Inverses and Radical Functions

Inverse Functions

Inverse functions reverse the input-output relationship of a function. If $f$ takes an input $a$ and produces $b$, then $f^{-1}$ takes $b$ and returns $a$. This concept is essential for solving equations, undoing operations, and working with radical functions that arise from "reversing" polynomials.

Graphing of Inverse Functions

The graph of $f^{-1}(x)$ is the reflection of $f(x)$ across the line $y = x$. Every point $(a, b)$ on $f(x)$ becomes $(b, a)$ on $f^{-1}(x)$.

This reflection also swaps domain and range: the domain of $f(x)$ becomes the range of $f^{-1}(x)$, and vice versa.

Common function-inverse pairs you'll see in this unit:

• Linear functions (e.g., $f(x) = 2x + 1$) have inverses that are also linear
• Quadratic functions (e.g., $f(x) = x^2$) have inverses that are square root functions, but only after restricting the domain (more on this below)
• Cubic functions (e.g., $f(x) = x^3$) have inverses that are cube root functions

Not every function has an inverse. A function needs to be one-to-one (each output comes from exactly one input) to have an inverse. You can check 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 on that domain.

This is why $f(x) = x^2$ on all real numbers fails. The horizontal line $y = 4$ hits the graph at both $x = 2$ and $x = -2$. To fix this, you restrict the domain to $x \geq 0$ (or $x \leq 0$), making the function one-to-one. Then the inverse is $f^{-1}(x) = \sqrt{x}$.

Monotonic functions (functions that are entirely increasing or entirely decreasing) always pass the horizontal line test, so they always have inverses.

Applications of Inverse and Radical Functions

Inverse and radical functions show up in practical problems more often than you might expect.

Unit conversion: The formula $f(x) = \frac{9}{5}x + 32$ converts Celsius to Fahrenheit. Its inverse, $f^{-1}(x) = \frac{5}{9}(x - 32)$, converts Fahrenheit back to Celsius. Any time you have a formula that converts one way, the inverse converts the other way.

Area and volume problems: If $f(x) = \sqrt{x}$ gives the side length of a square with area $x$, then $f^{-1}(x) = x^2$ gives the area from the side length. Similarly, $f(x) = \sqrt[3]{x}$ gives the edge length of a cube with volume $x$, and $f^{-1}(x) = x^3$ recovers the volume.

Exponential and logarithmic relationships: If $f(x) = 2^x$ models exponential growth, then $f^{-1}(x) = \log_2(x)$ lets you solve for the time needed to reach a given amount.

One useful algebraic fact: radical expressions can be rewritten with rational exponents (e.g., $\sqrt[3]{x} = x^{1/3}$), which often makes them easier to manipulate when finding inverses.

Domains of Radical Composite Functions

When you compose two functions, like $f(g(x))$, the domain depends on both pieces. The key rule for radicals:

• Even-indexed radicals (square root, fourth root, etc.) require the radicand to be $\geq 0$
• Odd-indexed radicals (cube root, fifth root, etc.) accept any real number

To find the domain of a radical composite function $f(g(x))$:

1. Identify the domain of the inner function $g(x)$
2. Determine which $x$-values make the radicand of the outer function non-negative
3. Take the intersection of those two sets

Worked example: Let $f(x) = \sqrt{x}$ and $g(x) = x^2 - 9$, so $f(g(x)) = \sqrt{x^2 - 9}$.

1. The domain of $g(x) = x^2 - 9$ is all real numbers.

2. The radicand must satisfy $x^2 - 9 \geq 0$, which factors as $(x-3)(x+3) \geq 0$. Testing intervals gives $x \leq -3$ or $x \geq 3$.

3. Intersecting with all real numbers changes nothing, so the domain is $(-\infty, -3] \cup [3, \infty)$.

Properties of Inverse Functions

A few properties worth committing to memory:

• If $f$ and $f^{-1}$ are true inverses, then $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This is the defining property and a reliable way to verify your answer.
• The domain and range swap between a function and its inverse.
• A function must be one-to-one to have an inverse. If it isn't naturally one-to-one, you can restrict its domain to make it one-to-one (as with $x^2$ restricted to $x \geq 0$).