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Math  /  Algebra

QuestionWhat is (f−g)(x)(f-g)(x) ? f(x)=−3x2−9x−6g(x)=x+2\begin{array}{l} f(x)=-3 x^{2}-9 x-6 \\ g(x)=x+2 \end{array}
Write your answer as a polynomial or a rational function in simplest form.

Studdy Solution

STEP 1

1. We are given two functions f(x) f(x) and g(x) g(x) .
2. We need to find the expression for (f−g)(x) (f-g)(x) .
3. The subtraction of functions involves subtracting their corresponding expressions.

STEP 2

1. Write the expression for (f−g)(x) (f-g)(x) .
2. Simplify the resulting expression.

STEP 3

Write the expression for (f−g)(x) (f-g)(x) by subtracting g(x) g(x) from f(x) f(x) :
(f−g)(x)=f(x)−g(x)=(−3x2−9x−6)−(x+2) (f-g)(x) = f(x) - g(x) = (-3x^2 - 9x - 6) - (x + 2)

STEP 4

Distribute the negative sign across the terms in g(x) g(x) :
(f−g)(x)=−3x2−9x−6−x−2 (f-g)(x) = -3x^2 - 9x - 6 - x - 2

STEP 5

Combine like terms:
- Combine the x x terms: −9x−x=−10x -9x - x = -10x - Combine the constant terms: −6−2=−8 -6 - 2 = -8
Thus, the expression simplifies to:
(f−g)(x)=−3x2−10x−8 (f-g)(x) = -3x^2 - 10x - 8
The simplest form of (f−g)(x) (f-g)(x) is:
−3x2−10x−8 \boxed{-3x^2 - 10x - 8}

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