User the function $f(x)=\log _{5}(x-3)-2$ to answer the following:
(a) Determine the equation of the asymptote of $f(x)$.
(b) Determine the domain of $f(x)$ in interval notation.
(c) Determine the range of $f(x)$ in interval notation.

Answer

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Answer

The range of $f(x)$ in interval notation is $\boxed{(-\infty, \infty)}$

Steps

Step 1 ：Set the argument of the logarithm equal to zero: $x-3=0$

Step 2 ：Solve for $x$ to get $x=3$

Step 3 ：$\boxed{x=3}$ is the equation of the asymptote of $f(x)$

Step 4 ：Set $x-3>0$ to find the domain of $f(x)$

Step 5 ：Solve for $x$ to get $x>3$

Step 6 ：The domain of $f(x)$ in interval notation is $\boxed{(3, \infty)}$

Step 7 ：Since the base of the logarithm is greater than 1, the function is increasing for all values in its domain

Step 8 ：Therefore, the function will take on all real number values

Step 9 ：The range of $f(x)$ in interval notation is $\boxed{(-\infty, \infty)}$