Question

let $f(x)=3x^2 - 5x - 8$ and $g(x)=x + 1$. perform the function operation and then find the domain of the result.
$\frac{f}{g}(x)$
$\frac{f}{g}(x)=\square$ (simplify your answer.)

Explanation:

Step1: Define the function quotient

$\frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{3x^2 - 5x - 8}{x + 1}$

Step2: Factor the numerator

Factor $3x^2 - 5x - 8$:
We find two numbers that multiply to $3\times(-8)=-24$ and add to $-5$, which are $-8$ and $3$.
Rewrite and factor:
$3x^2 + 3x - 8x - 8 = 3x(x+1) -8(x+1) = (3x - 8)(x + 1)$
Substitute back:
$\frac{(3x - 8)(x + 1)}{x + 1}$

Step3: Simplify the expression

Cancel the common factor $x+1$ (where $x
eq -1$):
$\frac{(3x - 8)\cancel{(x + 1)}}{\cancel{x + 1}} = 3x - 8$

Step4: Find the domain

The original function is undefined when $g(x)=0$. Solve $x+1=0$: $x=-1$.
So the domain is all real numbers except $x=-1$.

Answer:

$\frac{f}{g}(x)=3x - 8$
Domain: All real numbers where $x
eq -1$