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Math  /  Algebra

QuestionFind (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x). f(x)=−3x2−1,g(x)=2xf(x)=-3 x^{2}-1, g(x)=2 x (f∘g)(x)=(f \circ g)(x)= □\square (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given two functions f(x)=−3x2−1 f(x) = -3x^2 - 1 and g(x)=2x g(x) = 2x .
2. We need to find the composition of these functions in two different orders: (f∘g)(x) (f \circ g)(x) and (g∘f)(x) (g \circ f)(x) .
3. The 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) .
(f∘g)(x)=f(g(x))=f(2x) (f \circ g)(x) = f(g(x)) = f(2x)

STEP 4

Substitute 2x 2x into the function f(x)=−3x2−1 f(x) = -3x^2 - 1 .
f(2x)=−3(2x)2−1 f(2x) = -3(2x)^2 - 1

STEP 5

Simplify the expression:
(2x)2=4x2 (2x)^2 = 4x^2
f(2x)=−3(4x2)−1=−12x2−1 f(2x) = -3(4x^2) - 1 = -12x^2 - 1
Therefore, (f∘g)(x)=−12x2−1 (f \circ g)(x) = -12x^2 - 1 .

STEP 6

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(−3x2−1) (g \circ f)(x) = g(f(x)) = g(-3x^2 - 1)

STEP 7

Substitute −3x2−1 -3x^2 - 1 into the function g(x)=2x g(x) = 2x .
g(−3x2−1)=2(−3x2−1) g(-3x^2 - 1) = 2(-3x^2 - 1)

STEP 8

Simplify the expression:
g(−3x2−1)=2(−3x2)+2(−1)=−6x2−2 g(-3x^2 - 1) = 2(-3x^2) + 2(-1) = -6x^2 - 2
Therefore, (g∘f)(x)=−6x2−2 (g \circ f)(x) = -6x^2 - 2 .
The solutions are: (f∘g)(x)=−12x2−1 (f \circ g)(x) = -12x^2 - 1 (g∘f)(x)=−6x2−2 (g \circ f)(x) = -6x^2 - 2

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