QuestionFind $(f \circ g)(3)$ and $(g \circ f)(-2)$ for $f(x)=\sqrt{x+16}$ and $g(x)=x^{2}$ .

Studdy Solution

STEP 1

Assumptions1. We have two functions $f(x) = \sqrt{x+16}$ and $g(x) = x^{}$ . . We are asked to find the values of the compositions of these functions $f \circ g$ and $g \circ f$ .
3. The composition of functions $f \circ g$ means $f(g(x))$ , and $g \circ f$ means $g(f(x))$ .

STEP 2

First, let's find the value of $(f \circ g)()$ . This means we need to substitute $x =$ into $g(x)$ , and then substitute this result into $f(x)$ .
$f(g()) = f(^{2})$

STEP 3

Calculate the value of $g(3)$ .
$g(3) =3^{2} =9$

STEP 4

Now, substitute $g(3) =9$ into $f(x)$ .
$f(g(3)) = f(9) = \sqrt{9+16}$

STEP 5

Calculate the value of $\sqrt{9+16}$ .
$\sqrt{9+16} = \sqrt{25} =5$ So, $(f \circ g)(3) =5$ .

STEP 6

Next, let's find the value of $(g \circ f)(-2)$ . This means we need to substitute $x = -2$ into $f(x)$ , and then substitute this result into $g(x)$ .
$g(f(-2)) = g(\sqrt{-2+16})$

STEP 7

Calculate the value of $f(-2)$ .
$f(-2) = \sqrt{-2+16} = \sqrt{14}$

STEP 8

Now, substitute $f(-2) = \sqrt{14}$ into $g(x)$ .
$g(f(-2)) = g(\sqrt{14}) = (\sqrt{14})^{2}$

STEP 9

Calculate the value of $(\sqrt{14})^{2}$ .
$(\sqrt{14})^{2} =14$ So, $(g \circ f)(-2) =14$ .

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