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Math  /  Algebra

QuestionUse f(x)=x2−9f(x)=x^{2}-9 and g(x)=x2+9g(x)=x^{2}+9 to find a formula for each expression. Identify its domain. (a) (f+g)(x)(f+g)(x) (c) (fg)(x)(\mathrm{fg})(\mathrm{x}) (b) (f−g)(x)(f-g)(x) (d) (f/g)(x)(f / g)(x) (a) (f+g)(x)=(f+g)(x)= □\square (Simplify your answer. Do not factor.)

Studdy Solution

STEP 1

1. We are given two functions, f(x)=x2−9 f(x) = x^2 - 9 and g(x)=x2+9 g(x) = x^2 + 9 .
2. We need to find formulas for the expressions (f+g)(x) (f+g)(x) , (f−g)(x) (f-g)(x) , (fg)(x) (fg)(x) , and (f/g)(x) (f/g)(x) .
3. We need to identify the domain for each expression.
4. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

STEP 2

1. Calculate (f+g)(x) (f+g)(x) and identify its domain.
2. Calculate (f−g)(x) (f-g)(x) and identify its domain.
3. Calculate (fg)(x) (fg)(x) and identify its domain.
4. Calculate (f/g)(x) (f/g)(x) and identify its domain.

STEP 3

Calculate (f+g)(x) (f+g)(x) by adding the functions f(x) f(x) and g(x) g(x) :
(f+g)(x)=f(x)+g(x)=(x2−9)+(x2+9) (f+g)(x) = f(x) + g(x) = (x^2 - 9) + (x^2 + 9)
Combine like terms:
(f+g)(x)=x2−9+x2+9=2x2 (f+g)(x) = x^2 - 9 + x^2 + 9 = 2x^2
The domain of (f+g)(x) (f+g)(x) is all real numbers, since there are no restrictions on x x for a polynomial.

STEP 4

Calculate (f−g)(x) (f-g)(x) by subtracting the functions f(x) f(x) and g(x) g(x) :
(f−g)(x)=f(x)−g(x)=(x2−9)−(x2+9) (f-g)(x) = f(x) - g(x) = (x^2 - 9) - (x^2 + 9)
Combine like terms:
(f−g)(x)=x2−9−x2−9=−18 (f-g)(x) = x^2 - 9 - x^2 - 9 = -18
The domain of (f−g)(x) (f-g)(x) is all real numbers, since it is a constant function.

STEP 5

Calculate (fg)(x) (fg)(x) by multiplying the functions f(x) f(x) and g(x) g(x) :
(fg)(x)=f(x)⋅g(x)=(x2−9)(x2+9) (fg)(x) = f(x) \cdot g(x) = (x^2 - 9)(x^2 + 9)
Use the difference of squares formula:
(fg)(x)=(x2)2−(9)2=x4−81 (fg)(x) = (x^2)^2 - (9)^2 = x^4 - 81
The domain of (fg)(x) (fg)(x) is all real numbers, since it is a polynomial.

STEP 6

Calculate (f/g)(x) (f/g)(x) by dividing the functions f(x) f(x) and g(x) g(x) :
(f/g)(x)=f(x)g(x)=x2−9x2+9 (f/g)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 9}{x^2 + 9}
The domain of (f/g)(x) (f/g)(x) is all real numbers except where the denominator is zero. Since x2+9 x^2 + 9 is never zero for real x x , the domain is all real numbers.
The formulas and domains are as follows: (a) (f+g)(x)=2x2 (f+g)(x) = 2x^2 , Domain: all real numbers. (b) (f−g)(x)=−18 (f-g)(x) = -18 , Domain: all real numbers. (c) (fg)(x)=x4−81 (fg)(x) = x^4 - 81 , Domain: all real numbers. (d) (f/g)(x)=x2−9x2+9 (f/g)(x) = \frac{x^2 - 9}{x^2 + 9} , Domain: all real numbers.

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