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Math  /  Algebra

QuestionPoints: 0 of 1
If f(x)=xf(x)=\sqrt{x} and g(x)=−2x−2g(x)=-2 x-2, find (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x). (f∘g)(x)=(f \circ g)(x)= □\square (Simplify your answer. Type an exact answer, using radicals as needed.)

Studdy Solution

STEP 1

1. The function f(x)=x f(x) = \sqrt{x} is defined for x≥0 x \geq 0 .
2. The function g(x)=−2x−2 g(x) = -2x - 2 is a linear function defined for all real numbers.
3. The composition of functions (f∘g)(x) (f \circ g)(x) means f(g(x)) f(g(x)) .
4. The composition of functions (g∘f)(x) (g \circ f)(x) means g(f(x)) g(f(x)) .

STEP 2

1. Find (f∘g)(x) (f \circ g)(x) .
2. Find (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−2) (f \circ g)(x) = f(g(x)) = f(-2x - 2)
Since f(x)=x f(x) = \sqrt{x} , we have:
f(−2x−2)=−2x−2 f(-2x - 2) = \sqrt{-2x - 2}

STEP 4

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(\sqrt{x})
Since g(x)=−2x−2 g(x) = -2x - 2 , we have:
g(x)=−2(x)−2 g(\sqrt{x}) = -2(\sqrt{x}) - 2
The simplified compositions are: - (f∘g)(x)=−2x−2 (f \circ g)(x) = \sqrt{-2x - 2} - (g∘f)(x)=−2x−2 (g \circ f)(x) = -2\sqrt{x} - 2

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