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QuestionEvaluate (h⋅g)(−4)(h \cdot g)(-4) for the functions f(x)=−6xf(x)=-6x, g(x)=∣x+7∣g(x)=|x+7|, h(x)=1x+5h(x)=\frac{1}{x+5}. Answer: (h⋅g)(−4)(h \cdot g)(-4) is □\square.

Studdy Solution

STEP 1

Assumptions1. The function f(x)=−6xf(x)=-6x . The function g(x)=∣x+7∣g(x)=|x+7|
3. The function h(x)=1x+5h(x)=\frac{1}{x+5}
4. We are asked to evaluate the function (h⋅g)(−4)(h \cdot g)(-4)

STEP 2

The notation (h⋅g)(−4)(h \cdot g)(-4) means that we first apply the function gg to −4-4, and then apply the function hh to the result. In other words, we are looking for h(g(−4))h(g(-4)).

STEP 3

First, we need to find the value of g(−)g(-). We can do this by substituting −- into the function g(x)g(x).
g(−)=∣−+7∣g(-) = |-+7|

STEP 4

Calculate the value of g(−4)g(-4).
g(−4)=∣−4+7∣=∣3∣=3g(-4) = |-4+7| = |3| =3

STEP 5

Now that we have the value of g(−4)g(-4), we can substitute this into the function h(x)h(x) to find the value of h(g(−4))h(g(-4)).
h(g(−4))=h(3)=13+5h(g(-4)) = h(3) = \frac{1}{3+5}

STEP 6

Calculate the value of h(g(−4))h(g(-4)).
h(g(−4))=h(3)=13+5=18h(g(-4)) = h(3) = \frac{1}{3+5} = \frac{1}{8}So, (h⋅g)(−4)(h \cdot g)(-4) is 18\frac{1}{8}.

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