Functions
Quotient of two functions: Advanced
Kailee
-uppose that the functions $f$ and $g$ are defined as follows.
Español
\[
f(x)=\frac{x}{x-9} \quad g(x)=\frac{x+3}{x-9}
\]
Find $\frac{f}{g}$. Then, give its domain using an interval or union of intervals. Simplify your answers.
\[
\begin{array}{l}
\left(\frac{f}{g}\right)(x)=\square \\
\text { Domain of } \frac{f}{g}: \square
\end{array}
\]
Explanation
Check
Answer
The domain of the function is all real numbers except -3. In interval notation, this is: \(\boxed{(-\infty, -3) \cup (-3, \infty)}\)
Step 1 ：\(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\frac{x}{x-9}}{\frac{x+3}{x-9}}\)
Step 2 ：Multiply the numerator and the denominator by (x-9) to get rid of the denominator in both fractions: \(\left(\frac{f}{g}\right)(x) = \frac{x}{x+3}\)
Step 3 ：Set the denominator equal to zero and solve for x to find the domain: x + 3 = 0, x = -3
Step 4 ：The domain of the function is all real numbers except -3. In interval notation, this is: \(\boxed{(-\infty, -3) \cup (-3, \infty)}\)
