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Math  /  Algebra

QuestionGiven f(x)=x+17f(x)=x+17 and g(x)=5x2+3x−14g(x)=5 x^{2}+3 x-14, find (f−g)(−2)(f-g)(-2).

Studdy Solution

STEP 1

1. The functions f(x) f(x) and g(x) g(x) are defined as f(x)=x+17 f(x) = x + 17 and g(x)=5x2+3x−14 g(x) = 5x^2 + 3x - 14 .
2. We need to find the value of the function (f−g)(x) (f-g)(x) at x=−2 x = -2 .

STEP 2

1. Define the function (f−g)(x) (f-g)(x) .
2. Substitute x=−2 x = -2 into the function (f−g)(x) (f-g)(x) .
3. Calculate the result.

STEP 3

Define the function (f−g)(x) (f-g)(x) as the difference between f(x) f(x) and g(x) g(x) :
(f−g)(x)=f(x)−g(x) (f-g)(x) = f(x) - g(x) (f−g)(x)=(x+17)−(5x2+3x−14) (f-g)(x) = (x + 17) - (5x^2 + 3x - 14)

STEP 4

Simplify the expression for (f−g)(x) (f-g)(x) :
(f−g)(x)=x+17−5x2−3x+14 (f-g)(x) = x + 17 - 5x^2 - 3x + 14 (f−g)(x)=−5x2+(x−3x)+(17+14) (f-g)(x) = -5x^2 + (x - 3x) + (17 + 14) (f−g)(x)=−5x2−2x+31 (f-g)(x) = -5x^2 - 2x + 31

STEP 5

Substitute x=−2 x = -2 into the simplified expression for (f−g)(x) (f-g)(x) :
(f−g)(−2)=−5(−2)2−2(−2)+31 (f-g)(-2) = -5(-2)^2 - 2(-2) + 31

STEP 6

Calculate each term:
1. Calculate −5(−2)2 -5(-2)^2 : −5(−2)2=−5×4=−20 -5(-2)^2 = -5 \times 4 = -20
2. Calculate −2(−2) -2(-2) : −2(−2)=4 -2(-2) = 4
3. Sum the results: (f−g)(−2)=−20+4+31 (f-g)(-2) = -20 + 4 + 31

STEP 7

Add the values to find the final result:
(f−g)(−2)=−20+4+31=15 (f-g)(-2) = -20 + 4 + 31 = 15
The value of (f−g)(−2) (f-g)(-2) is:
15 \boxed{15}

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