Composite functions

Form composite functions f(g(x)) and g(f(x)), evaluate them, and find the domain and range of a composite.

One function after another

A feeds the output of one function into another. In $f(g(x))$, apply $g$ first, then apply $f$ to the result. It is also written $f \circ g$.

Example 1●

If $f(x) = x^2$ and $g(x) = x + 3$, find $f(g(2))$ and $g(f(2))$.

$f(g(2)) = f(5) = 25$

Inside first: g(2) = 5.

$g(f(2)) = g(4) = 7$

Now f first: f(2) = 4.

Example 2●●

With the same $f$ and $g$, find $f(g(x))$ and $g(f(x))$ as expressions.

$f(g(x)) = (x + 3)^2 = x^2 + 6x + 9$

Replace the x in f with the whole of g(x).

$g(f(x)) = x^2 + 3$

These are different, so order matters.

Different graphs: one is shifted left 3, the other up 3.

f(g(x)) versus g(f(x))

f squares, g adds a number. Doing g first shifts sideways; doing f first shifts up.

Try this

Is there any value of the added number for which the two graphs are the same?

For $f(g(x))$ to exist, $x$ must be in the domain of $g$, and $g(x)$ must be in the domain of $f$.

Example 3●●●

Let $f(x) = \sqrt{x}$ and $g(x) = 4 - x^2$. Find the domain and range of $f(g(x))$.

$f(g(x)) = \sqrt{4 - x^2}$

Need $4 - x^2 \ge 0$, so the domain is $-2 \le x \le 2$.

The output of g must be allowed into the square root.

Range $0 \le y \le 2$. It is the upper semicircle of radius $2$.

4 − x² runs from 0 to 4 on that domain, and √ of that runs from 0 to 2.

Flashcards

f(g(x))

Apply $g$ first, then $f$.

Is f(g(x)) = g(f(x))?

Usually not. Order matters.

Domain of f(g(x))

$x$ in the domain of $g$, with $g(x)$ in the domain of $f$.

Practice