Question

find each of the following functions and state their domains. (enter the domains in interval notation.)
$f(x)=x^{3}+5x^{2},\\ g(x)=7x^{2}-1$
(a) $f + g$
$f + g=$

domain

(b) $f - g$
$f - g=$

domain

(c) $fg$
$fg=$

domain

(d) $f/g$
$f/g=$

domain

Explanation:

Part (a): \( f + g \)

Step 1: Add the functions

To find \( f + g \), we add \( f(x) \) and \( g(x) \):
\( f(x) + g(x) = (x^3 + 5x^2) + (7x^2 - 1) \)
Combine like terms: \( x^3 + (5x^2 + 7x^2) - 1 = x^3 + 12x^2 - 1 \)

Step 2: Determine the domain

Both \( f(x) \) and \( g(x) \) are polynomials. The domain of a polynomial is all real numbers, so the domain of \( f + g \) is also all real numbers. In interval notation, that's \( (-\infty, \infty) \).

Step 1: Subtract the functions

To find \( f - g \), we subtract \( g(x) \) from \( f(x) \):
\( f(x) - g(x) = (x^3 + 5x^2) - (7x^2 - 1) \)
Distribute the negative sign: \( x^3 + 5x^2 - 7x^2 + 1 \)
Combine like terms: \( x^3 - 2x^2 + 1 \)

Step 2: Determine the domain

Both \( f(x) \) and \( g(x) \) are polynomials. The domain of a polynomial is all real numbers, so the domain of \( f - g \) is also all real numbers. In interval notation, that's \( (-\infty, \infty) \).

Step 1: Multiply the functions

To find \( fg \), we multiply \( f(x) \) and \( g(x) \):
\( f(x) \cdot g(x) = (x^3 + 5x^2)(7x^2 - 1) \)
Use the distributive property (FOIL for polynomials):
\( x^3(7x^2) + x^3(-1) + 5x^2(7x^2) + 5x^2(-1) \)
Simplify each term: \( 7x^5 - x^3 + 35x^4 - 5x^2 \)
Rearrange terms: \( 7x^5 + 35x^4 - x^3 - 5x^2 \)

Step 2: Determine the domain

Both \( f(x) \) and \( g(x) \) are polynomials. The domain of a polynomial is all real numbers, so the domain of \( fg \) is also all real numbers. In interval notation, that's \( (-\infty, \infty) \).

Answer:

\( f + g = \boldsymbol{x^3 + 12x^2 - 1} \)
domain: \( \boldsymbol{(-\infty, \infty)} \)

Part (b): \( f - g \)