Question

find f+g, f−g, fg, and \\(\frac{f}{g}\\). determine the domain for each function.\\(f(x)=x + 6\\), \\(g(x)=5x^2\\)\\(\left(\frac{f}{g}\
ight)(x)=\frac{x + 6}{5x^2}\\) (simplify your answer.)what is the domain of \\(\frac{f}{g}\\)?\\(\boldsymbol{\text{oa.}}\\) the domain of \\(\frac{f}{g}\\) is \\(\\{\\}\\). (use a comma to separate answers as needed.)\\(\boldsymbol{\text{ob.}}\\) the domain of \\(\frac{f}{g}\\) is \\((-\infty, 0)\cup(0, \infty)\\). (type your answer in interval notation.)\\(\boldsymbol{\text{oc.}}\\) the domain of \\(\frac{f}{g}\\) is \\(\varnothing\\).

Explanation:

Step1: Find $f+g$

$f(x)+g(x) = (x+6) + 5x^2 = 5x^2+x+6$
Domain: All real numbers, $(-\infty, \infty)$

Step2: Find $f-g$

$f(x)-g(x) = (x+6) - 5x^2 = -5x^2+x+6$
Domain: All real numbers, $(-\infty, \infty)$

Step3: Find $fg$

$f(x)\cdot g(x) = (x+6)\cdot 5x^2 = 5x^3+30x^2$
Domain: All real numbers, $(-\infty, \infty)$

Step4: Find $\frac{f}{g}$

$\frac{f(x)}{g(x)} = \frac{x+6}{5x^2}$
Domain: Exclude $x$ where $g(x)=0$. Solve $5x^2=0 \implies x=0$. So domain is $(-\infty,0)\cup(0,\infty)$

Answer:

1. $f+g = 5x^2+x+6$, Domain: $(-\infty, \infty)$
2. $f-g = -5x^2+x+6$, Domain: $(-\infty, \infty)$
3. $fg = 5x^3+30x^2$, Domain: $(-\infty, \infty)$
4. $\frac{f}{g} = \frac{x+6}{5x^2}$, Domain: $(-\infty,0)\cup(0,\infty)$
5. Correct domain for $\frac{f}{g}$: B. The domain of $\frac{f}{g}$ is $(-\infty,0)\cup(0,\infty)$