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Math  /  Algebra

Question(1 point) Given that f(x)=∣x∣f(x)=|x| and g(x)=7x+2g(x)=7 x+2, calculate (a) (f∘g)(x)=(f \circ g)(x)= □\square (b) (g∘f)(x)=(g \circ f)(x)= □\square (c) (g∘g)(x)=(g \circ g)(x)= □\square

Studdy Solution

STEP 1

1. We have two functions: f(x)=∣x∣ f(x) = |x| and g(x)=7x+2 g(x) = 7x + 2 .
2. We need to calculate the compositions of these functions: (f∘g)(x) (f \circ g)(x) , (g∘f)(x) (g \circ f)(x) , and (g∘g)(x) (g \circ g)(x) .

STEP 2

1. Calculate (f∘g)(x) (f \circ g)(x) .
2. Calculate (g∘f)(x) (g \circ f)(x) .
3. Calculate (g∘g)(x) (g \circ g)(x) .

STEP 3

To find (f∘g)(x) (f \circ g)(x) , substitute g(x) g(x) into f(x) f(x) :
(f∘g)(x)=f(g(x))=f(7x+2) (f \circ g)(x) = f(g(x)) = f(7x + 2)

STEP 4

Since f(x)=∣x∣ f(x) = |x| , we have:
f(7x+2)=∣7x+2∣ f(7x + 2) = |7x + 2|
Thus,
(f∘g)(x)=∣7x+2∣ (f \circ g)(x) = |7x + 2|

STEP 5

To find (g∘f)(x) (g \circ f)(x) , substitute f(x) f(x) into g(x) g(x) :
(g∘f)(x)=g(f(x))=g(∣x∣) (g \circ f)(x) = g(f(x)) = g(|x|)

STEP 6

Substitute into g(x)=7x+2 g(x) = 7x + 2 :
g(∣x∣)=7∣x∣+2 g(|x|) = 7|x| + 2
Thus,
(g∘f)(x)=7∣x∣+2 (g \circ f)(x) = 7|x| + 2

STEP 7

To find (g∘g)(x) (g \circ g)(x) , substitute g(x) g(x) into itself:
(g∘g)(x)=g(g(x))=g(7x+2) (g \circ g)(x) = g(g(x)) = g(7x + 2)

STEP 8

Substitute into g(x)=7x+2 g(x) = 7x + 2 :
g(7x+2)=7(7x+2)+2 g(7x + 2) = 7(7x + 2) + 2
Simplify:
=49x+14+2 = 49x + 14 + 2 =49x+16 = 49x + 16
Thus,
(g∘g)(x)=49x+16 (g \circ g)(x) = 49x + 16
The solutions are: (a) (f∘g)(x)=∣7x+2∣ (f \circ g)(x) = |7x + 2| (b) (g∘f)(x)=7∣x∣+2 (g \circ f)(x) = 7|x| + 2 (c) (g∘g)(x)=49x+16 (g \circ g)(x) = 49x + 16

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