Math Snap

STEP 1

1. We are given two functions, $f(x) = x^2 - 9$ and $g(x) = x^2 + 9$.
2. We need to find formulas for the expressions $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $(f/g)(x)$.
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 $(f+g)(x)$ and identify its domain.
2. Calculate $(f-g)(x)$ and identify its domain.
3. Calculate $(fg)(x)$ and identify its domain.
4. Calculate $(f/g)(x)$ and identify its domain.

STEP 3

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

STEP 4

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

STEP 5

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

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