Explanation
Finding the product of the given functions.
The product of two functions is defined as follows:
[tex](f\cdot g)(x)=(f)(x)\cdot(g)(x)[/tex]
Then, we have:
[tex]\begin{gathered} f(x)=\frac{6}{x+7} \\ g(x)=x+5 \\ (f\cdot g)(x)=(f)(x)*(g)(x) \\ (f\cdot g)(x)=\frac{6}{x+7}(x+5) \\ (f\cdot g)(x)=\frac{6(x+5)}{x+7} \\ (f\cdot g)(x)=\frac{6x+30}{x+7} \end{gathered}[/tex]
Finding the domain of f · g
Step 1: We set the denominator equal to 0 to find where the above expression is undefined.
[tex]x+7=0[/tex]
Step 2: We subtract 7 from both sides.
[tex]\begin{gathered} x+7-7=0-7 \\ x=-7 \end{gathered}[/tex]
Step 3: Since the domain is all values of x that make the expression defined, the domain of f · g is all values of x different from -7.
[tex]\begin{gathered} D={}{}\lbrace x|x\ne-7\rbrace \\ \text{ or} \\ D=(-\infty,-7)\cup(-7,\infty) \end{gathered}[/tex]Answer[tex]D=(-\infty,-7)\cup(-7,\infty)[/tex]
