Functions and Function Notation

Function Notation

$f(x) = 3x + 2$ means: input $x$, multiply by 3, add 2.

$f(4) = 3(4) + 2 = 14$.

Composite Functions

$fg(x) = f(g(x))$: apply $g$ first, then $f$.

If $f(x) = 2x + 1$ and $g(x) = x^2$: $fg(3) = f(g(3)) = f(9) = 19$ $gf(3) = g(f(3)) = g(7) = 49$

Note: $fg(x) \neq gf(x)$ in general.

Inverse Functions

$f^{-1}(x)$ reverses $f$. To find it:

1. Write $y = f(x)$.
2. Swap $x$ and $y$.
3. Rearrange for $y$.

$f(x) = 3x + 2$. Let $y = 3x + 2$. Swap: $x = 3y + 2$. Rearrange: $y = \frac{x-2}{3}$.

$f^{-1}(x) = \frac{x-2}{3}$.

Domain and Range

• Domain: set of allowed inputs.
• Range: set of possible outputs.