Before you read

Answer first — see what you already know.

If $f (x) = x + 5$ and $g (x) = 2 x$ , what is $f (g (3))$ ?

Check your answer

Answer:

Before you read

Answer first — see what you already know.

If $f (x) = x^{2}$ and $g (x) = x + 1$ , what is $g (f (x))$ ?

Check your answer

Answer:

Theory

Function Composition

$(f \circ g)(x) = f(g(x))$ : plug $g(x)$ into $f$ .

Important: $f(g(x)) \neq g(f(x))$ in general! Order matters.

Example: $f(x) = x^2$ , $g(x) = x + 3$ .
- $f(g(x)) = f(x+3) = (x+3)^2$
- $g(f(x)) = g(x^2) = x^2 + 3$

Different results!

Finding the inverse function of f(x) = 2x + 3

Example 1

If $f(x) = 2x + 1$ and $g(x) = x - 4$ , find $f(g(3))$ .

$g(3) = 3 - 4 = -1$

$f(-1) = 2(-1) + 1 = -1$

f(g(3)) = -1

Theory

Inverse Functions

The inverse $f^{-1}(x)$ undoes $f$ : $f(f^{-1}(x)) = x$ .

To find $f^{-1}$ :
1. Write $y = f(x)$
2. Swap $x$ and $y$
3. Solve for $y$

Key property: The graph of $f^{-1}$ is the reflection of $f$ across $y = x$ .

If $(a, b)$ is on $f$ , then $(b, a)$ is on $f^{-1}$ .

Example 1

Find the inverse of $f(x) = 3x - 6$ .

$y = 3x - 6$

Swap: $x = 3y - 6$

Solve: $x + 6 = 3y$ , $y = \frac{x + 6}{3}$

f^{-1}(x) = \frac{x + 6}{3}

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