Question

let $f(x)=4x^2 - 29x + 30$ and $g(x)=x - 6$. 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 operation

$\frac{f}{g}(x) = \frac{4x^2 - 29x + 30}{x - 6}$

Step2: Factor the numerator

Factor $4x^2 - 29x + 30$: find two numbers that multiply to $4*30=120$ and add to $-29$, which are $-24$ and $-5$.
$4x^2 -24x -5x +30 = 4x(x-6) -5(x-6) = (4x-5)(x-6)$
So $\frac{f}{g}(x) = \frac{(4x-5)(x-6)}{x - 6}$

Step3: Simplify the expression

Cancel the common factor $(x-6)$ (where $x
eq 6$):
$\frac{f}{g}(x) = 4x - 5$

Step4: Determine the domain

The original function is undefined when $g(x)=0$, so $x-6=0 \implies x=6$. The domain is all real numbers except $x=6$.

Answer:

$\frac{f}{g}(x) = 4x - 5$
Domain: All real numbers except $x=6$, or in interval notation $(-\infty, 6) \cup (6, \infty)$