Composite and Inverse Functions

In advanced math, functions can interact in various ways. Two of the most important concepts are composite functions (combining two functions) and inverse functions (reversing a function).

Composite Functions

Composing two functions means plugging the output of one function directly into the input of another. The composition of $f$ and $g$ is denoted by $(f \circ g)(x)$ and is defined as:

$(f \circ g)(x) = f(g(x))$

This tells you to evaluate $g(x)$ first, and then plug that result into $f(x)$.

Example 1: Evaluating at a specific number If $f(x) = 2x + 1$ and $g(x) = x^2$, find $(f \circ g)(3)$.

1. First, find $g(3)$: $g(3) = 3^2 = 9$.
2. Next, plug this result into $f$: $f(9) = 2(9) + 1 = 18 + 1 = 19$. So, $(f \circ g)(3) = 19$.

Example 2: Finding a composite function Using the same functions, find $(g \circ f)(x)$. Here, we plug the entire function $f(x)$ into $g(x)$: $(g \circ f)(x) = g(f(x)) = g(2x + 1)$ $= (2x + 1)^2 = 4x^2 + 4x + 1$

Inverse Functions

An inverse function, denoted as $f^{-1}(x)$, reverses the input-output relationship of the original function $f(x)$. If $f(x)$ turns an $x$ into a $y$, then $f^{-1}(x)$ turns that $y$ back into the original $x$.

For a function to have an inverse, it must be one-to-one, meaning every output is paired with exactly one input (it passes the horizontal line test).

How to find an inverse function:

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

Example 3: Finding an inverse function Find the inverse of $f(x) = \frac{3x - 1}{x + 2}$.

1. Replace $f(x)$ with $y$: $y = \frac{3x - 1}{x + 2}$
2. Swap $x$ and $y$: $x = \frac{3y - 1}{y + 2}$
3. Solve for $y$: Multiply both sides by $(y + 2)$: $x(y + 2) = 3y - 1$ Distribute the $x$: $xy + 2x = 3y - 1$ Move all terms containing $y$ to one side: $xy - 3y = -2x - 1$ Factor out $y$: $y(x - 3) = -2x - 1$ Divide by $(x - 3)$: $y = \frac{-2x - 1}{x - 3}$
4. Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \frac{-2x - 1}{x - 3}$