Lesson 10.1: Finding Composite and Inverse Functions

New Concept

This lesson introduces function composition, where one function's output is another's input. We'll use this to define, find, and verify inverse functions—special functions that 'undo' the original, a key concept for our upcoming topics.

What’s next

Now, you'll apply these definitions. Get ready for interactive examples and practice cards on composing functions, identifying one-to-one functions, and finding their inverses.

Property

The composition of functions f and g is written $f \circ g$ and is defined by

We read $f(g(x))$ as $f$ of $g$ of $x$. In composition, the output of one function is the input of a second function.

Examples

• For $f(x) = 5x + 1$ and $g(x) = x - 3$, find $(f \circ g)(x)$. We compute $f(g(x)) = f(x-3) = 5(x-3) + 1 = 5x - 15 + 1 = 5x - 14$.

• For $f(x) = 5x + 1$ and $g(x) = x - 3$, find $(g \circ f)(x)$. We compute $g(f(x)) = g(5x+1) = (5x+1) - 3 = 5x - 2$. Note that $(f \circ g)(x) \neq (g \circ f)(x)$.

Property

A function is one-to-one if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each $y$-value is matched with only one $x$-value. There are no repeated $y$-values.

Horizontal Line Test
If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.

Examples

• The set {(1, 2), (3, 4), (5, 6)} is a one-to-one function because each $y$-value (2, 4, 6) is paired with only one $x$-value.

Property

If $f(x)$ is a one-to-one function whose ordered pairs are of the form $(x, y)$, then its inverse function $f^{-1}(x)$ is the set of ordered pairs $(y, x)$. The domain of $f$ is the range of $f^{-1}$ and the domain of $f^{-1}$ is the range of $f$. The graphs of $f$ and $f^{-1}$ are mirror images of each other through the line $y=x$.

Examples

• The inverse of the function {(0, 1), (2, 5), (4, 9)} is the function {(1, 0), (5, 2), (9, 4)}.

• If the function $f$ has domain {0, 2, 4} and range {1, 5, 9}, its inverse $f^{-1}$ has domain {1, 5, 9} and range {0, 2, 4}.

Property

Two functions $f(x)$ and $g(x)$ are inverses of each other if both of the following conditions are true for all $x$ in the respective domains:

Examples

• Let $f(x) = 2x+3$ and $g(x) = \frac{x-3}{2}$. To verify, we find $f(g(x)) = 2(\frac{x-3}{2}) + 3 = (x-3) + 3 = x$.

• Continuing the previous example, we must also check the other direction: $g(f(x)) = \frac{(2x+3)-3}{2} = \frac{2x}{2} = x$. Since both are true, they are inverses.

Property

To find the inverse of a one-to-one function algebraically:
Step 1. Substitute $y$ for $f(x)$.
Step 2. Interchange the variables $x$ and $y$.
Step 3. Solve for $y$.
Step 4. Substitute $f^{-1}(x)$ for $y$.

Examples

• Find the inverse of $f(x) = 3x - 5$. Step 1: $y = 3x - 5$. Step 2: $x = 3y - 5$. Step 3: $x+5 = 3y$, so $y = \frac{x+5}{3}$. Step 4: $f^{-1}(x) = \frac{x+5}{3}$.

• Find the inverse of $f(x) = \sqrt[3]{x-1}$. Step 1: $y = \sqrt[3]{x-1}$. Step 2: $x = \sqrt[3]{y-1}$. Step 3: $x^3 = y-1$, so $y = x^3+1$. Step 4: $f^{-1}(x) = x^3+1$.