Composite functions

A composite function feeds the output of one function into another.
The notation $f(g(x))$ means "put $g(x)$ in place of the input of $f$".
You always work from the inside out: deal with the inner function first, then the outer one.

To evaluate at a number, find the inner value, then apply the outer function to it.
Suppose $f(2) = 4$ and $g(x) = 2^x + 1$.
Then $g(f(2)) = g(4) = 2^4 + 1 = 17$.

The same works when the functions have formulas.
Let $f(x) = ax + 2$ and $g(x) = 3x - 1$.
To find $f(g(-1))$, first $g(-1) = 3(-1) - 1 = -4$, then $f(-4) = a(-4) + 2$.

You can also compose a function with itself and simplify symbolically.
For $h(x) = 4 - 2x$, the composite $h(h(x))$ replaces the input of $h$ with $h(x)$: $h(h(x)) = 4 - 2(h(x)) = 4 - 2(4 - 2x)$.
Multiplying out gives $4 - 8 + 4x = 4x - 4$.

Finally, the order of composition matters.
In general $f(g(x))$ is not the same as $g(f(x))$, because putting $g$ inside $f$ is different from putting $f$ inside $g$.

Let $f(x) = 2x + 1$ and $g(x) = x^2$. What is $f(g(3))$?
Work inside first: $g(3) = 3^2 = 9$.
Then $f(9) = 2(9) + 1 = 19$.

Let $f(x) = 3x$ and $g(x) = x + 4$. What is $g(f(2))$?
Work inside first: $f(2) = 3(2) = 6$.
Then $g(6) = 6 + 4 = 10$.

Let $p(x) = 3x - 1$. What is $p(p(x))$?
Replace the input of $p$ with $p(x)$: $p(p(x)) = 3(3x - 1) - 1$.
Multiplying out gives $9x - 3 - 1 = 9x - 4$.