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Math  /  Algebra

QuestionConsider the following piecewise-defined function. f(x)={7−x−4−3 If x<−4−36x+8+7 if x>−4f(x)=\left\{\begin{array}{ll} 7^{-x-4}-3 & \text { If } x<-4 \\ \frac{-36}{x+8}+7 & \text { if } x>-4 \end{array}\right.
Step 3 of 3 : Evaluate this function at x=−6x=-6. Write the exact answer. Do not round. If the answer is undefined, write Und as your answer. f(−6)=□f(-6)=\square

Studdy Solution

STEP 1

1. We have a piecewise-defined function f(x) f(x) .
2. The function has two cases: - f(x)=7−x−4−3 f(x) = 7^{-x-4} - 3 for x<−4 x < -4 - f(x)=−36x+8+7 f(x) = \frac{-36}{x+8} + 7 for x>−4 x > -4
3. We need to evaluate f(x) f(x) at x=−6 x = -6 .

STEP 2

1. Determine which case of the piecewise function applies for x=−6 x = -6 .
2. Substitute x=−6 x = -6 into the appropriate expression.
3. Simplify the expression to find the exact value of f(−6) f(-6) .

STEP 3

Determine which case applies for x=−6 x = -6 .
Since −6<−4 -6 < -4 , we use the first case of the piecewise function:
f(x)=7−x−4−3 f(x) = 7^{-x-4} - 3

STEP 4

Substitute x=−6 x = -6 into the expression 7−x−4−3 7^{-x-4} - 3 .
f(−6)=7−(−6)−4−3 f(-6) = 7^{-(-6)-4} - 3

STEP 5

Simplify the exponent:
f(−6)=76−4−3 f(-6) = 7^{6-4} - 3 f(−6)=72−3 f(-6) = 7^{2} - 3

STEP 6

Calculate 72 7^2 and then subtract 3:
f(−6)=49−3 f(-6) = 49 - 3 f(−6)=46 f(-6) = 46
The exact value of f(−6) f(-6) is:
46 \boxed{46}

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