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 $f(kx)$.

$f(g(x))$

$f(x) = x$

$f(x) + k$

$f(x) - k$

$f(kx)$

$f(kx)$

$f(g(x))$: 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.