Suppose that the functions $f$ and $g$ are defined as follows. $\newline$ $f(x)=\frac{x}{x-9} \quad g(x)=\frac{8}{x+5}$ $\newline$ Find $\frac{f}{g}$ . Then, give its domain using an interval or union of intervals. Simplify your answers. $\newline$ $\left(\frac{f}{g}\right)(x)=\prod$ $\newline$ Domain of $\frac{f}{g}$ :

Full solution

Q. Suppose that the functions

f

and

g

are defined as follows.

\newline

f(x)=\frac{x}{x-9} \quad g(x)=\frac{8}{x+5}

\newline

Find

\frac{f}{g}

. Then, give its domain using an interval or union of intervals. Simplify your answers.

\newline

\left(\frac{f}{g}\right)(x)=\prod

\newline

Domain of

\frac{f}{g}

Define functions and quotient: Step $1$ : Define the functions and set up the quotient. $\newline$ $f(x) = \frac{x}{x-9}$ , $g(x) = \frac{8}{x+5}$ . $\newline$ To find $\frac{f}{g}(x)$ , we need to divide $f(x)$ by $g(x)$ : $\newline$ $\left(\frac{f}{g}\right)(x) = \frac{\frac{x}{x-9}}{\frac{8}{x+5}}$ .
Simplify expression: Step $2$ : Simplify the expression for $(f/g)(x)$ . $\newline$ Using the property of division of fractions, $(a/b) / (c/d) = (a\cdot d)/(b\cdot c)$ , we get: $\newline$ $((f)/(g))(x) = (x/(x-9)) \cdot ((x+5)/8) = (x\cdot(x+5))/(8\cdot(x-9))$ .
Numerator and denominator: Step $3$ : Simplify the numerator and the denominator. $\newline$ The numerator $x*(x+5)$ expands to $x^2 + 5x$ . $\newline$ So, $\left(\frac{f}{g}\right)(x) = \frac{x^2 + 5x}{8*(x-9)}$ .
Determine domain: Step $4$ : Determine the domain of $(f/g)(x)$ . The domain is all real numbers except where the denominator is zero. For $f(x)$ , $x \neq 9$ (since $x-9 \neq 0$ ). For $g(x)$ , $x \neq -5$ (since $x+5 \neq 0$ ). Thus, the domain of $(f/g)(x)$ is all real numbers except $x = 9$ and $x = -5$ .