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Math  /  Algebra

QuestionLet⁡f(x)=3x2−2x\operatorname{Let}_{f(x)}=3 x^{2}-2 x  o 3x2−2x−1\text { o } 3 x^{2}-2 x-1
Operate Me...
1. (f−g)(x)(f-g)(x)
2. (f+h)(x)(f+h)(x) - 2x2−2x2 x^{2}-2 x
3. g(x)h(x)\frac{g(x)}{h(x)} - x3x\frac{x}{3 x}
4. (h−g)(x)(h-g)(x) - 2x2−2x−22 x^{2}-2 x-2
5. f(−1)⋅g(2)⋅h(0)f(-1) \cdot g(2) \cdot h(0) - 24 - 7x2−4x−37 x^{2}-4 x-3 00−240_{0}-24

Studdy Solution

STEP 1

1. f(x)=3x2−2x f(x) = 3x^2 - 2x
2. g(x)=3x2−2x−1 g(x) = 3x^2 - 2x - 1
3. h(x)=2x2−2x h(x) = 2x^2 - 2x
4. We need to find the expressions and values for each given operation.

STEP 2

1. Compute (f−g)(x)(f - g)(x)
2. Compute (f+h)(x)(f + h)(x)
3. Compute g(x)h(x)\frac{g(x)}{h(x)}
4. Compute (h−g)(x)(h - g)(x)
5. Compute f(−1)⋅g(2)⋅h(0)f(-1) \cdot g(2) \cdot h(0)

STEP 3

Compute (f−g)(x)(f - g)(x): (f−g)(x)=f(x)−g(x)(f - g)(x) = f(x) - g(x) (f−g)(x)=(3x2−2x)−(3x2−2x−1)(f - g)(x) = (3x^2 - 2x) - (3x^2 - 2x - 1) (f−g)(x)=3x2−2x−3x2+2x+1(f - g)(x) = 3x^2 - 2x - 3x^2 + 2x + 1 (f−g)(x)=1(f - g)(x) = 1

STEP 4

Compute (f+h)(x)(f + h)(x): (f+h)(x)=f(x)+h(x)(f + h)(x) = f(x) + h(x) (f+h)(x)=(3x2−2x)+(2x2−2x)(f + h)(x) = (3x^2 - 2x) + (2x^2 - 2x) (f+h)(x)=3x2−2x+2x2−2x(f + h)(x) = 3x^2 - 2x + 2x^2 - 2x (f+h)(x)=5x2−4x(f + h)(x) = 5x^2 - 4x

STEP 5

Compute g(x)h(x)\frac{g(x)}{h(x)}: g(x)h(x)=3x2−2x−12x2−2x\frac{g(x)}{h(x)} = \frac{3x^2 - 2x - 1}{2x^2 - 2x} =(3x2−2x−1)2x(x−1)= \frac{(3x^2 - 2x - 1)}{2x(x - 1)} This expression cannot be simplified further.

STEP 6

Compute (h−g)(x)(h - g)(x): (h−g)(x)=h(x)−g(x)(h - g)(x) = h(x) - g(x) (h−g)(x)=(2x2−2x)−(3x2−2x−1)(h - g)(x) = (2x^2 - 2x) - (3x^2 - 2x - 1) (h−g)(x)=2x2−2x−3x2+2x+1(h - g)(x) = 2x^2 - 2x - 3x^2 + 2x + 1 (h−g)(x)=−x2+1(h - g)(x) = -x^2 + 1

STEP 7

Compute f(−1)⋅g(2)⋅h(0)f(-1) \cdot g(2) \cdot h(0): f(−1)=3(−1)2−2(−1)=3+2=5f(-1) = 3(-1)^2 - 2(-1) = 3 + 2 = 5 g(2)=3(2)2−2(2)−1=12−4−1=7g(2) = 3(2)^2 - 2(2) - 1 = 12 - 4 - 1 = 7 h(0)=2(0)2−2(0)=0h(0) = 2(0)^2 - 2(0) = 0 f(−1)⋅g(2)⋅h(0)=5⋅7⋅0=0f(-1) \cdot g(2) \cdot h(0) = 5 \cdot 7 \cdot 0 = 0

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