$f(g(x))$: Apply $g$ first, then $f$. | $g(f(x))$: Apply $f$ first, then $g$. The results are generally different.

Vertical Translation: Shifts the graph up or down. | Horizontal Dilation: Stretches or shrinks the graph horizontally.

Evaluate $g(x)$ first, then use the result as the input for $f(x)$. Work from the inside out.

Composition is generally not commutative; $f(g(x))$ is usually not equal to $g(f(x))$.

The result is the original function; $f(x) = x$ implies $g(f(x)) = f(g(x)) = g(x)$.

Substitute the entire function $g(x)$ for every instance of $x$ in $f(x)$.

They can represent transformations such as vertical translations ($f(x) + k$) and horizontal dilations ($f(kx)$).

Find the output of $g(x)$ from its graph, then use that output as the input for $f(x)$ on its graph.

Replace $x$ in $f(x)$ with $g(x)$: $f(g(x)) = (x^2) + 1 = x^2 + 1$.

First, find $g(2) = 2^2 + 3 = 7$. Then, find $f(7) = 2(7) - 1 = 13$.

Let $g(x) = x + 2$ and $f(x) = x^2$. Then $f(g(x)) = (x + 2)^2 = h(x)$.

First find $f(g(x)) = 2x + 1$. Then solve $2x + 1 = 5$, which gives $x = 2$.

Substitute $g(x)$ into $f(x)$: $f(g(x)) = (\sqrt{x})^2 = x$, for $x \geq 0$.

Find the domain of $g(x)$ and ensure that the range of $g(x)$ is within the domain of $f(x)$.