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Math  /  Algebra

Question5. If f(x)=−x2−2x+1f(x)=-x^{2}-2 x+1 and g(x)=x+2g(x)=x+2 a. Find (fog)(f o g) b. Find g(f(−2)g(f(-2)

Studdy Solution

STEP 1

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

STEP 2

1. Find the composition (f∘g)(x) (f \circ g)(x) .
2. Evaluate f(−2) f(-2) .
3. Use the result from Step 2 to find g(f(−2)) g(f(-2)) .

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(x+2) (f \circ g)(x) = f(g(x)) = f(x + 2)

STEP 4

Substitute x+2 x + 2 into f(x) f(x) :
f(x+2)=−(x+2)2−2(x+2)+1 f(x + 2) = -(x + 2)^2 - 2(x + 2) + 1

STEP 5

Expand and simplify the expression:
f(x+2)=−(x2+4x+4)−2x−4+1 f(x + 2) = -(x^2 + 4x + 4) - 2x - 4 + 1 f(x+2)=−x2−4x−4−2x−4+1 f(x + 2) = -x^2 - 4x - 4 - 2x - 4 + 1 f(x+2)=−x2−6x−7 f(x + 2) = -x^2 - 6x - 7
So, (f∘g)(x)=−x2−6x−7 (f \circ g)(x) = -x^2 - 6x - 7 .

STEP 6

Evaluate f(−2) f(-2) by substituting −2-2 into f(x) f(x) :
f(−2)=−(−2)2−2(−2)+1 f(-2) = -(-2)^2 - 2(-2) + 1

STEP 7

Simplify the expression:
f(−2)=−(4)+4+1 f(-2) = -(4) + 4 + 1 f(−2)=−4+4+1 f(-2) = -4 + 4 + 1 f(−2)=1 f(-2) = 1

STEP 8

Use the result from Step 5 to find g(f(−2)) g(f(-2)) :
Since f(−2)=1 f(-2) = 1 , substitute 1 1 into g(x) g(x) :
g(1)=1+2 g(1) = 1 + 2

STEP 9

Simplify the expression:
g(1)=3 g(1) = 3
The composition (f∘g)(x) (f \circ g)(x) is:
−x2−6x−7 \boxed{-x^2 - 6x - 7}
The value of g(f(−2)) g(f(-2)) is:
3 \boxed{3}

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