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Math  /  Algebra

QuestionIf f(x)=x2−6x+2f(x)=x^{2}-6 x+2 and g(x)=−2xg(x)=-2 x, find the following composition. (g∘f)(2)(g∘f)(2)=□\begin{array}{r} (g \circ f)(2) \\ (g \circ f)(2)=\square \end{array}

Studdy Solution

STEP 1

1. We are given two functions: f(x)=x2−6x+2 f(x) = x^2 - 6x + 2 and g(x)=−2x g(x) = -2x .
2. We need to find the composition (g∘f)(2) (g \circ f)(2) , which means g(f(2)) g(f(2)) .

STEP 2

1. Evaluate the function f(x) f(x) at x=2 x = 2 .
2. Use the result from Step 1 as the input for the function g(x) g(x) .
3. Evaluate g(x) g(x) using the result from Step 1.

STEP 3

Substitute x=2 x = 2 into the function f(x) f(x) :
f(2)=(2)2−6(2)+2 f(2) = (2)^2 - 6(2) + 2

STEP 4

Simplify the expression:
f(2)=4−12+2 f(2) = 4 - 12 + 2 f(2)=−6 f(2) = -6

STEP 5

Use the result from Step 1, f(2)=−6 f(2) = -6 , as the input for g(x) g(x) :
g(f(2))=g(−6) g(f(2)) = g(-6)

STEP 6

Substitute x=−6 x = -6 into the function g(x) g(x) :
g(−6)=−2(−6) g(-6) = -2(-6)

STEP 7

Simplify the expression:
g(−6)=12 g(-6) = 12
The value of the composition (g∘f)(2) (g \circ f)(2) is:
12 \boxed{12}

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