3.8 Inverses and Radical Functions

Inverse Functions

Inverse functions reverse what the original function does. If a function takes an input and produces an output, the inverse takes that output and returns the original input. This makes them essential for solving equations where you need to work backward, and they connect directly to radical functions, which are the inverses of power functions.

Graphically, a function and its inverse are reflections of each other across the line $y = x$. This visual relationship is a quick way to check your work and understand how domains and ranges swap between a function and its inverse.

more resources to help you study

practice questions

Solving for Inverse Functions

Two functions $f(x)$ and $g(x)$ are inverses if they undo each other completely. The formal test: $f(g(x)) = x$ and $g(f(x)) = x$. Both compositions must equal $x$ for the functions to be true inverses.

To find the inverse of a function algebraically:

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

For polynomial functions, step 3 often requires factoring or the quadratic formula. For example, finding the inverse of $f(x) = 2x^3 + 1$: swap to get $x = 2y^3 + 1$, then solve to get $y = \sqrt[3]{\frac{x-1}{2}}$.

For rational functions, the same steps apply, but you'll typically need to collect all $y$-terms on one side and factor $y$ out. Watch for domain restrictions: values that make a denominator zero in the original function become excluded from the range of the inverse, and vice versa.

Graphing Inverse Functions

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

• The domain of the inverse is the range of the original, and the range of the inverse is the domain of the original
• The x-intercepts of the original become y-intercepts of the inverse, and vice versa
• If the original function is not one-to-one (fails the horizontal line test), you must restrict its domain before finding an inverse. For instance, $f(x) = x^2$ is restricted to $x \geq 0$ so that $f^{-1}(x) = \sqrt{x}$ is a valid function

One common misconception: inverse functions do not automatically share the same symmetry properties as the original. An even function (symmetric about the y-axis) cannot have an inverse over its full domain precisely because it fails the horizontal line test. Once you restrict the domain, the symmetry is broken.

Applications of Inverse Functions

Radical functions are the inverses of power functions. That's why they show up so often in this unit.

• The inverse of $f(x) = x^2$ (with $x \geq 0$) is $f^{-1}(x) = \sqrt{x}$
• The inverse of $f(x) = x^3$ is $f^{-1}(x) = \sqrt[3]{x}$ (no domain restriction needed since cubics are already one-to-one)

Sphere radius example: The volume of a sphere is $V = \frac{4}{3}\pi r^3$. If you know the volume and need the radius, solve for $r$ by applying the inverse:

$r = \sqrt[3]{\frac{3V}{4\pi}}$

So a sphere with volume $100 \text{ cm}^3$ has radius $r = \sqrt[3]{\frac{3(100)}{4\pi}} \approx 2.88 \text{ cm}$.

Projectile example: If the height of an object is $h(t) = -16t^2 + 64t + 5$, finding the time when it reaches a specific height means setting $h(t)$ equal to that height and solving the resulting quadratic. This is using the inverse relationship, though the full inverse function requires restricting the domain (since the parabola is not one-to-one). You'd typically choose the portion of the domain that makes physical sense (e.g., $0 \leq t \leq 2$ for the upward phase).

Special Types of Functions and Their Inverses

• Exponential and logarithmic functions are inverses of each other: $f(x) = b^x$ and $f^{-1}(x) = \log_b(x)$. You'll work with these extensively in later units, but the inverse relationship follows the same reflection-across-$y = x$ principle.
• Piecewise functions can have inverses, but you need to verify that each piece is one-to-one and that the pieces together cover the range without overlap. Find the inverse of each piece separately, then assemble them with the appropriate domain restrictions.