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Math  /  Algebra

QuestionPrevious Problem Problem List Next Problem (1 point) Express the function y=4(x−7)5y=4(x-7)^{5} as a composition y=(f∘g)(x)y=(f \circ g)(x) of two simpler functions. f(x)=f(x)= □\square g(x)=g(x)= □\square Help entering formulas After submitting an answer, you can get a hint and click "Show Me Another" to see a similar probler

Studdy Solution

STEP 1

1. We are given the function y=4(x−7)5 y = 4(x-7)^5 .
2. We need to express this function as a composition of two simpler functions y=(f∘g)(x) y = (f \circ g)(x) .

STEP 2

1. Identify the inner function g(x) g(x) .
2. Identify the outer function f(x) f(x) .
3. Verify the composition (f∘g)(x) (f \circ g)(x) matches the original function.

STEP 3

Identify the inner function g(x) g(x) .
The inner function is the expression inside the parentheses, which is:
g(x)=x−7 g(x) = x - 7

STEP 4

Identify the outer function f(x) f(x) .
The outer function takes the result of g(x) g(x) and applies the remaining operations. In this case, it raises the input to the 5th power and multiplies by 4:
f(x)=4x5 f(x) = 4x^5

STEP 5

Verify the composition (f∘g)(x) (f \circ g)(x) .
Substitute g(x)=x−7 g(x) = x - 7 into f(x) f(x) :
(f∘g)(x)=f(g(x))=f(x−7)=4(x−7)5 (f \circ g)(x) = f(g(x)) = f(x - 7) = 4(x - 7)^5
This matches the original function y=4(x−7)5 y = 4(x - 7)^5 .
The functions are:
f(x)=4x5 f(x) = 4x^5 g(x)=x−7 g(x) = x - 7

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