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Math  /  Algebra

QuestionEvaluate the function f(x)=x2+4x−4f(x)=x^{2}+4 x-4 at the given values of the independent variable and simplify a. f(−2)f(-2) b. f(x+7)f(x+7) c. f(−x)f(-x) a. f(−2)=f(-2)= □\square (Simplify your answer.) b. f(x+7)=f(x+7)= □\square (Simplify your answer) c. f(−x)=□f(-x)=\square □\square (Simplify your answer)

Studdy Solution

STEP 1

1. The function f(x)=x2+4x−4f(x) = x^2 + 4x - 4 is a quadratic function.
2. To evaluate the function at specific values, we substitute the given values into the function and simplify.
3. We will substitute −2-2, x+7x+7, and −x-x into the function separately for each part.

STEP 2

1. Evaluate f(−2)f(-2).
2. Evaluate f(x+7)f(x+7).
3. Evaluate f(−x)f(-x).

STEP 3

Substitute x=−2x = -2 into the function f(x)f(x).
f(−2)=(−2)2+4(−2)−4 f(-2) = (-2)^2 + 4(-2) - 4

STEP 4

Simplify the expression.
f(−2)=4−8−4=−8 f(-2) = 4 - 8 - 4 = -8

STEP 5

Substitute x=x+7x = x+7 into the function f(x)f(x).
f(x+7)=(x+7)2+4(x+7)−4 f(x+7) = (x+7)^2 + 4(x+7) - 4

STEP 6

Expand the quadratic term (x+7)2(x+7)^2.
(x+7)2=x2+14x+49 (x+7)^2 = x^2 + 14x + 49

STEP 7

Substitute the expanded quadratic term back into the function f(x+7)f(x+7).
f(x+7)=x2+14x+49+4(x+7)−4 f(x+7) = x^2 + 14x + 49 + 4(x+7) - 4

STEP 8

Distribute and combine like terms.
f(x+7)=x2+14x+49+4x+28−4=x2+18x+73 f(x+7) = x^2 + 14x + 49 + 4x + 28 - 4 = x^2 + 18x + 73

STEP 9

Substitute x=−xx = -x into the function f(x)f(x).
f(−x)=(−x)2+4(−x)−4 f(-x) = (-x)^2 + 4(-x) - 4

STEP 10

Simplify the expression.
f(−x)=x2−4x−4 f(-x) = x^2 - 4x - 4
Solution: a. f(−2)=−8f(-2) = -8 b. f(x+7)=x2+18x+73f(x+7) = x^2 + 18x + 73 c. f(−x)=x2−4x−4f(-x) = x^2 - 4x - 4

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