$f(g(x))$: 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.

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 $f(kx)$.

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 $f(x)$.