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Math  /  Algebra

QuestionQuestion 10, 10.1.19 HW Score: 60\%, 12 of 20 Part 2 of 2 points Points: 0 of 1
If f(x)=x3+x+1f(x)=x^{3}+x+1 and g(x)=−4xg(x)=-4 x, find (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x). (f∘g)(x)=−64x3−4x+1(g∘f)(x)=□\begin{array}{l} (f \circ g)(x)=-64 x^{3}-4 x+1 \\ (g \circ f)(x)=\square \end{array}

Studdy Solution

STEP 1

1. We are given two functions f(x)=x3+x+1 f(x) = x^3 + x + 1 and g(x)=−4x g(x) = -4x .
2. We need to find the compositions (f∘g)(x) (f \circ g)(x) and (g∘f)(x) (g \circ f)(x) .
3. Composition of functions involves substituting one function into another.

STEP 2

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

STEP 3

To find (f∘g)(x) (f \circ g)(x) , substitute g(x) g(x) into f(x) f(x) . This means replacing every x x in f(x) f(x) with g(x)=−4x g(x) = -4x .
f(g(x))=f(−4x)=(−4x)3+(−4x)+1 f(g(x)) = f(-4x) = (-4x)^3 + (-4x) + 1

STEP 4

Calculate each term in the expression:
(−4x)3=−64x3 (-4x)^3 = -64x^3
Therefore, the expression becomes:
f(g(x))=−64x3−4x+1 f(g(x)) = -64x^3 - 4x + 1
This matches the given result for (f∘g)(x) (f \circ g)(x) .

STEP 5

To find (g∘f)(x) (g \circ f)(x) , substitute f(x) f(x) into g(x) g(x) . This means replacing every x x in g(x) g(x) with f(x)=x3+x+1 f(x) = x^3 + x + 1 .
g(f(x))=g(x3+x+1)=−4(x3+x+1) g(f(x)) = g(x^3 + x + 1) = -4(x^3 + x + 1)

STEP 6

Distribute the −4-4 across the terms inside the parentheses:
g(f(x))=−4(x3)−4(x)−4(1) g(f(x)) = -4(x^3) - 4(x) - 4(1)
g(f(x))=−4x3−4x−4 g(f(x)) = -4x^3 - 4x - 4
The composition (g∘f)(x) (g \circ f)(x) is −4x3−4x−4 \boxed{-4x^3 - 4x - 4} .

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