Composite and Inverse of Functions

Summary

In this topic, you will learn how to use function notation to describe simple functions, their inverses, and how to form composite functions.

• Composite Function — a function made by combining two functions, where the output of one function becomes the input of another. Example: If $f(x) = x^2$ and $g(x) = x + 2$, then the composite function $h(x) = f(g(x)) = (x + 2)^2$.
• Inverse Function — a function that reverses the effect of the original function, denoted as $f^{-1}(x)$. Example: If $f(x) = 2x + 1$, then the inverse function $f^{-1}(x)$ is found by solving $y = 2x + 1$ for $x$, giving $f^{-1}(x) = \frac{x - 1}{2}$.
• Self-Inverse Function — a function that is its own inverse. Example: If $f(x) = \frac{1}{x}$, then $f(f(x)) = x$, so $f$ is self-inverse.

Exam Tips

Key Definitions to Remember

• A composite function is a combination of two functions, written as $f(g(x))$.
• An inverse function reverses the original function, denoted as $f^{-1}(x)$.

Common Confusions

• Mixing up the order of functions in composite functions, $f(g(x))$ is not the same as $g(f(x))$.
• Assuming all functions have inverses; only one-to-one functions have inverses.

Typical Exam Questions

• What is $f^{-1}(x)$ for $f(x) = 3x + 5$? Solve $y = 3x + 5$ for $x$, giving $f^{-1}(x) = \frac{x - 5}{3}$.
• Find $g(f(x))$ if $f(x) = x^2$ and $g(x) = \sqrt{x}$. $g(f(x)) = \sqrt{x^2} = |x|$.
• Solve $2f(x) = g(x)$ for given functions. Substitute and solve the equation.

What Examiners Usually Test

• Understanding and applying the concept of composite functions.
• Finding inverse functions and verifying them.
• Solving equations involving composite and inverse functions.