Given the functions:
[tex]\begin{gathered} f(x)=\frac{5}{x+7} \\ \text{AND} \\ g(x)=\frac{2}{x} \end{gathered}[/tex]
Let's solve for the following:
• (a) f/g
To solve for f/g let's divide f(x) by g(x).
We have:
[tex]\frac{f}{g}=\frac{f(x)}{g(x)}=(\frac{f}{g})(x)=\frac{\frac{5}{x+7}}{\frac{2}{x}}[/tex]
Solving further, we have:
[tex]\begin{gathered} (\frac{f}{g})(x)=\frac{5}{x+7}\ast\frac{x}{2} \\ \\ (\frac{f}{g})(x)=\frac{5x}{2(x+7)} \end{gathered}[/tex]
Therefore, the function f/g is:
[tex](\frac{f}{g})(x)=\frac{5x}{2(x+7)}[/tex]
• (b) Domain of f/g.
The domain is the set of all possible x-values where the function is defined.
To find the domain, set the denominator to zero and solve for x.
We have:
[tex]2(x+7)=0[/tex]
Divide both sides by 2:
[tex]\begin{gathered} \frac{2(x+7)}{2}=\frac{0}{2} \\ \\ (x+7)=0 \end{gathered}[/tex]
Subtract 7 from both sides:
[tex]\begin{gathered} x+7-7=0-7 \\ \\ x=-7 \end{gathered}[/tex]
Therefore, the domian is:
[tex]\mleft(-\infty,-7\mright)\cup(-7,\infty)[/tex]
ANSWER:
[tex](a)\text{ ( }\frac{f}{g})(x)=\frac{5x}{2(x+7)}[/tex][tex](b)\text{ Domain: }(-\infty,-7)\cup(-7,\infty)[/tex]
