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QuestionEvaluate the following using f(x)=7x−1f(x)=7x-1 and g(x)=∣x∣g(x)=|x|: (a) (f∘g)(−5)(f \circ g)(-5) (b) (g∘f)(3)(g \circ f)(3) Find (f∘g)(−5)=(f \circ g)(-5)= (Simplify your answer.)

Studdy Solution

STEP 1

Assumptions1. We have two functions, f(x)=7x−1f(x)=7x-1 and g(x)=∣x∣g(x)=|x|. . We need to evaluate the compositions of these functions, namely (f∘g)(−5)(f \circ g)(-5) and (g∘f)(3)(g \circ f)(3).

STEP 2

The composition of two functions, (f∘g)(x)(f \circ g)(x), is defined as f(g(x))f(g(x)). This means that we first apply the function gg to xx, and then apply the function ff to the result.

STEP 3

To evaluate (f∘g)(−5)(f \circ g)(-5), we first apply the function gg to −5-5.g(−5)=∣−5∣g(-5) = |-5|

STEP 4

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

STEP 5

Now, we apply the function ff to the result from step4.
f(g(−5))=f(5)=7⋅5−1f(g(-5)) = f(5) =7 \cdot5 -1

STEP 6

Calculate the value of ⋅5−1 \cdot5 -1.
f(g(−5))=⋅5−1=34f(g(-5)) = \cdot5 -1 =34So, (f∘g)(−5)=34(f \circ g)(-5) =34.

STEP 7

Now, let's evaluate (g∘f)(3)(g \circ f)(3). This means we first apply the function ff to 33.
f(3)=7⋅3−1f(3) =7 \cdot3 -1

STEP 8

Calculate the value of 7⋅3−17 \cdot3 -1.
f(3)=7⋅3−1=20f(3) =7 \cdot3 -1 =20

STEP 9

Now, we apply the function gg to the result from step8.
g(f(3))=g(20)=∣20∣g(f(3)) = g(20) = |20|

STEP 10

Calculate the absolute value of 2020.
g(f(3))=∣20∣=20g(f(3)) = |20| =20So, (g∘f)(3)=20(g \circ f)(3) =20.

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