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Math  /  Algebra

QuestionGiven the following functions: f(x)=3x2+3x−6g(x)=3x+6\begin{array}{l} f(x)=3 x^{2}+3 x-6 \\ g(x)=3 x+6 \end{array}
Find each of the values below. Give exact answers. a. (f+g)(−2)=(f+g)(-2)= □\square b. (f−g)(3)=(f-g)(3)= □\square 15 c. (f⋅g)(2)=(f \cdot g)(2)= □\square 120 d. (fg)(4)=\left(\frac{f}{g}\right)(4)= □\square

Studdy Solution

STEP 1

1. We have two functions: f(x)=3x2+3x−6 f(x) = 3x^2 + 3x - 6 and g(x)=3x+6 g(x) = 3x + 6 .
2. We need to find the values of (f+g)(−2)(f+g)(-2), (f−g)(3)(f-g)(3), (f⋅g)(2)(f \cdot g)(2), and (fg)(4)\left(\frac{f}{g}\right)(4).

STEP 2

1. Calculate (f+g)(−2)(f+g)(-2).
2. Calculate (f−g)(3)(f-g)(3).
3. Calculate (f⋅g)(2)(f \cdot g)(2).
4. Calculate (fg)(4)\left(\frac{f}{g}\right)(4).

STEP 3

Calculate (f+g)(x)(f+g)(x) by adding the functions:
(f+g)(x)=f(x)+g(x) (f+g)(x) = f(x) + g(x) =(3x2+3x−6)+(3x+6) = (3x^2 + 3x - 6) + (3x + 6)

STEP 4

Simplify the expression:
(f+g)(x)=3x2+3x−6+3x+6 (f+g)(x) = 3x^2 + 3x - 6 + 3x + 6 =3x2+6x = 3x^2 + 6x

STEP 5

Substitute x=−2 x = -2 into (f+g)(x)(f+g)(x):
(f+g)(−2)=3(−2)2+6(−2) (f+g)(-2) = 3(-2)^2 + 6(-2)

STEP 6

Calculate the result:
(f+g)(−2)=3(4)−12 (f+g)(-2) = 3(4) - 12 =12−12 = 12 - 12 =0 = 0

STEP 7

Calculate (f−g)(x)(f-g)(x) by subtracting the functions:
(f−g)(x)=f(x)−g(x) (f-g)(x) = f(x) - g(x) =(3x2+3x−6)−(3x+6) = (3x^2 + 3x - 6) - (3x + 6)

STEP 8

Simplify the expression:
(f−g)(x)=3x2+3x−6−3x−6 (f-g)(x) = 3x^2 + 3x - 6 - 3x - 6 =3x2−12 = 3x^2 - 12

STEP 9

Substitute x=3 x = 3 into (f−g)(x)(f-g)(x):
(f−g)(3)=3(3)2−12 (f-g)(3) = 3(3)^2 - 12

STEP 10

Calculate the result:
(f−g)(3)=3(9)−12 (f-g)(3) = 3(9) - 12 =27−12 = 27 - 12 =15 = 15

STEP 11

Calculate (f⋅g)(x)(f \cdot g)(x) by multiplying the functions:
(f⋅g)(x)=f(x)⋅g(x) (f \cdot g)(x) = f(x) \cdot g(x) =(3x2+3x−6)(3x+6) = (3x^2 + 3x - 6)(3x + 6)

STEP 12

Substitute x=2 x = 2 into both functions:
f(2)=3(2)2+3(2)−6 f(2) = 3(2)^2 + 3(2) - 6 =3(4)+6−6 = 3(4) + 6 - 6 =12+6−6 = 12 + 6 - 6 =12 = 12
g(2)=3(2)+6 g(2) = 3(2) + 6 =6+6 = 6 + 6 =12 = 12

STEP 13

Calculate the result:
(f⋅g)(2)=f(2)⋅g(2) (f \cdot g)(2) = f(2) \cdot g(2) =12⋅12 = 12 \cdot 12 =144 = 144

STEP 14

Calculate (fg)(x)\left(\frac{f}{g}\right)(x) by dividing the functions:
(fg)(x)=f(x)g(x) \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}

STEP 15

Substitute x=4 x = 4 into both functions:
f(4)=3(4)2+3(4)−6 f(4) = 3(4)^2 + 3(4) - 6 =3(16)+12−6 = 3(16) + 12 - 6 =48+12−6 = 48 + 12 - 6 =54 = 54
g(4)=3(4)+6 g(4) = 3(4) + 6 =12+6 = 12 + 6 =18 = 18

STEP 16

Calculate the result:
(fg)(4)=f(4)g(4) \left(\frac{f}{g}\right)(4) = \frac{f(4)}{g(4)} =5418 = \frac{54}{18} =3 = 3
The values are: a. (f+g)(−2)=0(f+g)(-2) = 0 b. (f−g)(3)=15(f-g)(3) = 15 c. (f⋅g)(2)=144(f \cdot g)(2) = 144 d. (fg)(4)=3\left(\frac{f}{g}\right)(4) = 3

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