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Math  /  Algebra

QuestionA function is shown below. h(x)={−12x−15 for x≤−420−3x2 for x>−4h(x)=\left\{\begin{array}{ll} -\frac{1}{2} x-15 & \text { for } x \leq-4 \\ 20-3 x^{2} & \text { for } x>-4 \end{array}\right.
What is the value of h(−4)+3h(−2)h(-4)+3 h(-2) ?

Studdy Solution

STEP 1

1. We are given a piecewise function h(x) h(x) .
2. The function has two parts: h(x)=−12x−15 h(x) = -\frac{1}{2}x - 15 for x≤−4 x \leq -4 and h(x)=20−3x2 h(x) = 20 - 3x^2 for x>−4 x > -4 .
3. We need to find the value of h(−4)+3h(−2) h(-4) + 3h(-2) .

STEP 2

1. Determine which part of the piecewise function to use for h(−4) h(-4) .
2. Calculate h(−4) h(-4) .
3. Determine which part of the piecewise function to use for h(−2) h(-2) .
4. Calculate h(−2) h(-2) .
5. Compute h(−4)+3h(−2) h(-4) + 3h(-2) .

STEP 3

Determine the correct part of the piecewise function for h(−4) h(-4) :
Since −4≤−4 -4 \leq -4 , use h(x)=−12x−15 h(x) = -\frac{1}{2}x - 15 .

STEP 4

Calculate h(−4) h(-4) :
h(−4)=−12(−4)−15 h(-4) = -\frac{1}{2}(-4) - 15
Simplify:
h(−4)=2−15 h(-4) = 2 - 15 h(−4)=−13 h(-4) = -13

STEP 5

Determine the correct part of the piecewise function for h(−2) h(-2) :
Since −2>−4 -2 > -4 , use h(x)=20−3x2 h(x) = 20 - 3x^2 .

STEP 6

Calculate h(−2) h(-2) :
h(−2)=20−3(−2)2 h(-2) = 20 - 3(-2)^2
Simplify:
h(−2)=20−3(4) h(-2) = 20 - 3(4) h(−2)=20−12 h(-2) = 20 - 12 h(−2)=8 h(-2) = 8

STEP 7

Compute h(−4)+3h(−2) h(-4) + 3h(-2) :
h(−4)+3h(−2)=−13+3(8) h(-4) + 3h(-2) = -13 + 3(8)
Simplify:
h(−4)+3h(−2)=−13+24 h(-4) + 3h(-2) = -13 + 24 h(−4)+3h(−2)=11 h(-4) + 3h(-2) = 11
The value of h(−4)+3h(−2) h(-4) + 3h(-2) is:
11 \boxed{11}

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