Suppose that the functions $f$ and $g$ are defined as follows. $\newline$ $f(x)=\frac{9}{x} \quad g(x)=\frac{4}{x+3}$ $\newline$ Find $\frac{g}{f}$ . Then, give its domain using an interval or union of intervals. $\newline$ Simplify your answers. $\newline$ $\left(\frac{g}{f}\right)(x)=\frac{4x}{9(x+3)}$ $\newline$ Domain of $\frac{g}{f}$ : $(-\infty,-3) \cup (-3,\infty)$ $\newline$ $-\infty$

Full solution

Q. Suppose that the functions

f

and

g

are defined as follows.

\newline

f(x)=\frac{9}{x} \quad g(x)=\frac{4}{x+3}

\newline

Find

\frac{g}{f}

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

\newline

Simplify your answers.

\newline

\left(\frac{g}{f}\right)(x)=\frac{4x}{9(x+3)}

\newline

Domain of

\frac{g}{f}

(-\infty,-3) \cup (-3,\infty)

\newline

-\infty

Express Function Division: Express the function $(g/f)(x)$ as the division of $g(x)$ by $f(x)$ .
Given Functions: Write down the functions $f(x)$ and $g(x)$ as given: $f(x) = \frac{9}{x}$ and $g(x) = \frac{4}{x+3}$ .
Calculate $(g/f)(x)$ : Divide $g(x)$ by $f(x)$ to find $(g/f)(x)$ : $(g/f)(x) = \frac{g(x)}{f(x)} = \frac{\frac{4}{(x+3)}}{\frac{9}{x}}$ .
Simplify Division: Multiply by the reciprocal of the denominator to simplify the division: $(g/f)(x) = \left(\frac{4}{x+3}\right) \cdot \left(\frac{x}{9}\right)$ .
Multiply Numerators: Simplify the expression by multiplying the numerators and denominators: $(\frac{g}{f})(x) = \frac{4x}{9(x+3)}$ .
Identify Domain: Identify the domain of g/f)(x)\. The domain is all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for \$x: $9(x+3) = 0$ .
Solve for x: Solve the equation $9(x+3) = 0$ for $x$ : $x+3 = 0$ , so $x = -3$ .
Domain of $(g/f)(x)$ : The domain of $(g/f)(x)$ is all real numbers except $x = -3$ . In interval notation, this is $(-\infty, -3) \cup (-3, \infty)$ .