$f(g(x))$: 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.

A function formed by applying one function to the results of another: $f(g(x))$.

Apply $g$ to $x$ first, then apply $f$ to the result.

The function $f(x) = x$, which returns the input unchanged.

Breaking down a complex function into simpler component functions.

Shifting the graph of a function up or down by adding a constant, represented by $f(x) + k$.

Stretching or shrinking the graph of a function horizontally by multiplying the input by a constant, represented by $f(kx)$.