QuestionEvaluate $(f \circ g)(6)$ and $(g \circ f)(-3)$ for $f(x)=\sqrt{x+28}$ and $g(x)=x^{2}$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=\sqrt{x+28}$ . The function $g(x)$ is defined as $g(x)=x^{}$
3. The composition of functions is defined as $(f \circ g)(x) = f(g(x))$ and $(g \circ f)(x) = g(f(x))$

STEP 2

First, we need to find the value of $(f \circ g)(6)$ . This means we need to substitute $6$ into $g(x)$ and then substitute the result into $f(x)$ .
$(f \circ g)(6) = f(g(6))$

STEP 3

Substitute $6$ into $g(x)$ .
$g(6) =6^{2}$

STEP 4

Calculate the value of $g(6)$ .
$g(6) =6^{2} =36$

STEP 5

Now, substitute $g()$ into $f(x)$ .
$f(g()) = f(36)$

STEP 6

Substitute $36$ into $f(x)$ .
$f(36) = \sqrt{36+28}$

STEP 7

Calculate the value of $f(36)$ .
$f(36) = \sqrt{36+28} = \sqrt{64} =$ So, $(f \circ g)(6) =$ .

STEP 8

Next, we need to find the value of $(g \circ f)(-3)$ . This means we need to substitute $-3$ into $f(x)$ and then substitute the result into $g(x)$ .
$(g \circ f)(-3) = g(f(-3))$

STEP 9

Substitute $-3$ into $f(x)$ .
$f(-3) = \sqrt{-3+28}$

STEP 10

Calculate the value of $f(-3)$ .
$f(-3) = \sqrt{-3+28} = \sqrt{25} =5$

STEP 11

Now, substitute $f(-3)$ into $g(x)$ .
$g(f(-3)) = g(5)$

STEP 12

Substitute $5$ into $g(x)$ .
$g(5) =5^{2}$

STEP 13

Calculate the value of $g(5)$ .
$g(5) =5^{2} =25$ So, $(g \circ f)(-3) =25$ .

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