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QuestionEvaluate the following using f(x)=7x−3f(x)=7x-3 and g(x)=∣x∣g(x)=|x|: (a) (f∘g)(−4)(f \circ g)(-4) (b) (g∘f)(6)(g \circ f)(6) Find (f∘g)(−4)=(f \circ g)(-4)= (Simplify your answer.)

Studdy Solution

STEP 1

Assumptions1. We have two functions, f(x)=7x−3f(x)=7x-3 and g(x)=∣x∣g(x)=|x|. . We need to evaluate the compositions of these functions at specific points.

STEP 2

The composition of functions, denoted as (f∘g)(x)(f \circ g)(x), means to apply function gg first and then apply function ff to the result. So, (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)).

STEP 3

To find (f∘g)(−)(f \circ g)(-), we first apply the function gg to −-.g(−)=∣−∣g(-) = |-|

STEP 4

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

STEP 5

Now, we apply the function ff to the result g(−4)=4g(-4) =4.
f(g(−4))=f(4)=7⋅4−3f(g(-4)) = f(4) =7 \cdot4 -3

STEP 6

Calculate the value of f(g(−4))f(g(-4)).
f(g(−4))=⋅4−3=28−3=25f(g(-4)) = \cdot4 -3 =28 -3 =25So, (f∘g)(−4)=25(f \circ g)(-4) =25.

STEP 7

Now, let's find (g∘f)(6)(g \circ f)(6). The composition (g∘f)(x)(g \circ f)(x) means to apply function ff first and then apply function gg to the result. So, (g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x)).

STEP 8

To find (g∘f)(6)(g \circ f)(6), we first apply the function ff to 66.f(6)=7⋅6−3f(6) =7 \cdot6 -3

STEP 9

Calculate the value of f(6)f(6).
f(6)=7⋅6−3=42−3=39f(6) =7 \cdot6 -3 =42 -3 =39

STEP 10

Now, we apply the function gg to the result f(6)=39f(6) =39.
g(f(6))=g(39)=∣39∣g(f(6)) = g(39) = |39|

STEP 11

Calculate the value of g(f(6))g(f(6)).
g(f(6))=∣39∣=39g(f(6)) = |39| =39So, (g∘f)(6)=39(g \circ f)(6) =39.

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