TBIL-Precal Combining and Composing Functions (FN5)

In Activity 2.5.1 and Activity 2.5.2, we have found the sum, difference, product, and quotient of two functions. We can use the following notation for these newly created functions:

\begin{aligned} (f + g) (x) & = f (x) + g (x) \\ (f - g) (x) & = f (x) - g (x) \\ (f \cdot g) (x) & = f (x) \cdot g (x) \\ (\frac{f}{g}) (x) & = \frac{f (x)}{g (x)} \end{aligned}

🔗

With

(\frac{f}{g}) (x),

we note that the quotient is only defined when

g (x) \neq 0 .

🔗

Activity 2.5.4.

🔗

Let

f (x) = \frac{1}{3 x - 5} .

🔗

(a)

🔗

Find

f (4) .

$\frac{4}{3 x - 5}$
$\frac{1}{4 (3 x - 5)}$
$\frac{1}{7}$
$7$

Hint.

See Remark 2.2.5 for a reminder of what this notation means!

Answer.

🔗

(b)

🔗

If you were asked to find

f (x^{3} - 2),

how do you think you would proceed?

Multiply the original function $\frac{1}{3 x - 5}$ by $x^{3} - 2 .$
Plug the expression $x^{3} - 2$ in for all the $x$ -values in $\frac{1}{3 x - 5} .$
Plug the original function $\frac{1}{3 x - 5}$ in for all the $x$ -values in $x^{3} - 2 .$
Multiply $3 x - 5$ by $x^{3} - 2 .$

Answer.

🔗

(c)

🔗

Find

f (x^{3} - 2) .

$\frac{1}{3 x - 5} \cdot (x^{3} - 2)$
$\frac{1}{3 (x^{3} - 2) - 5}$
${(\frac{1}{3 x - 5})}^{3} - 2$
$(3 x - 5) (x^{3} - 2)$

Answer.

🔗

(d)

🔗

What if we gave the expression

x^{3} - 2

a name? Let’s define

g (x) = x^{3} - 2 .

What’s another way we could denote

f (x^{3} - 2) ?

$f (x) \cdot g (x)$
$g (f (x))$
$f (g (x))$
$\frac{f (x)}{g (x)}$

Answer.

🔗

Definition 2.5.5.

🔗

Given the functions

f (x)

and

g (x),

we define the composition of

f

and

g

to be the new function

h (x)

given by

h (x) = f (g (x)) .

🔗

We also sometimes use the notation

f \circ g

🔗

(f \circ g) (x)

🔗

to refer to

f (g (x)) .

🔗

Remark 2.5.6.

🔗

When discussing the composite function

f (g (x)),

also written as

(f \circ g) (x),

we often call

g (x)

the "inner function" and

f (x)

the "outer function". It is important to note that the inner function is actually the first function that gets applied to a given input, and then the outer function is applied to the output of the inner function.

🔗

Activity 2.5.7.

🔗

Let

f (x) = \frac{1}{3 x - 5}

and

g (x) = x^{3} - 2 .

🔗

(a)

🔗

Find

f (g (x)) .

$\frac{x^{3} - 2}{3 x - 5}$
$\frac{1}{(3 x - 5) (x^{3} - 2)}$
$\frac{1}{3 (x^{3} - 2) - 5}$
${(\frac{1}{3 x - 5})}^{3} - 2$

Answer.

🔗

(b)

🔗

Find

g (f (x)) .

$\frac{x^{3} - 2}{3 x - 5}$
$\frac{1}{(3 x - 5) (x^{3} - 2)}$
$\frac{1}{3 (x^{3} - 2) - 5}$
${(\frac{1}{3 x - 5})}^{3} - 2$

Answer.

🔗

Remark 2.5.8.

🔗

We can also evaluate the composition of two functions at a particular value just as we did with one function. For example, we may be asked to find something like

f (g (2))

(g \circ f) (- 3) .

🔗

Activity 2.5.9.

🔗

Let

f (x) = 2 x^{3}

and

g (x) = \sqrt{6 - x} .

🔗

(a)

🔗

Find

f (g (2)) .

$14$
$16$
$18$
$20$
undefined

Answer.

🔗

(b)

🔗

Find

(g \circ f) (- 3) .

$50$
$54$
$\sqrt{60}$
$\sqrt{- 48}$
undefined

Answer.

🔗

(c)

🔗

Find

(f \circ g) (10) .

$2 (\sqrt{- 4})^{3}$
$16$
$\sqrt{- 1994}$
$- 16$
undefined

Answer.

🔗

Remark 2.5.10.

🔗

As we saw in Activity 2.5.9, in order for a composite function to make sense, we need to ensure that the range of the inner function lies within the domain of the outer function so that the resulting composite function is defined at every possible input.

🔗

Remark 2.5.11.

🔗

In addition to the possibility that functions are given by formulas, functions can be given by tables or graphs. We can think about composite functions in these settings as well, and the following activities prompt us to consider functions given in this way.

🔗