Suppose that the functions $\newline$ $f$ and $\newline$ $g$ are defined as follows. $\newline$ $f(x)=\frac{8}{x}$ $\quad$ $g(x)=\frac{3}{x+9}$ $\newline$ Find $\newline$ $\frac{g}{f}$ . Then, give its domain using an interval or union of intervals. Simplify your answers. $\newline$ $\left(\frac{g}{f}\right)(x)=\left[\right.$

Full solution

Q. Suppose that the functions

\newline

f

and

\newline

g

are defined as follows.

\newline

f(x)=\frac{8}{x}

\quad

g(x)=\frac{3}{x+9}

\newline

Find

\newline

\frac{g}{f}

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

\newline

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

Divide Functions: To find the quotient $(g/f)(x)$ , we need to divide the function $g(x)$ by the function $f(x)$ . This means we will take the function $g(x) = \frac{3}{x+9}$ and divide it by $f(x) = \frac{8}{x}$ .
Multiply Reciprocals: The division of the two functions is performed by multiplying $g(x)$ by the reciprocal of $f(x)$ . So, $(g/f)(x) = \frac{3}{(x+9)} \times \frac{x}{8}$ .
Simplify Expression: Now we simplify the expression by multiplying the numerators and denominators: $(\frac{3x}{8(x+9)})$ .
Find Domain: The simplified form of $(g/f)(x)$ is $\frac{3x}{8(x+9)}$ . Now we need to find the domain of this function. The domain is all the values of $x$ for which the function is defined.
Identify Undefined Value: To find the domain, we look for values of $x$ that would make the denominator equal to zero, since division by zero is undefined. The denominator is $8(x+9)$ , which is zero when $x+9 = 0$ , or $x = -9$ .
Determine Interval Notation: Therefore, the domain of $(g/f)(x)$ is all real numbers except $x = -9$ . In interval notation, this is written as $(-\infty, -9) \cup (-9, \infty)$ .