Problem
Sketch the graph of the following function. Indicate where the function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. \[ f(x)=\frac{-1}{x-7} \] On what interval(s) is $\mathrm{f}$ increasing and on what interval(s) is $\mathrm{f}$ decreasing? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function is increasing on $\square$. The function is never decreasing. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function is decreasing on $\square$. The function is never increasing. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) C. The function is increasing on $\square$ and decreasing on (Simplify your answers. Type your answers in interval notation. Type exact answers, using radicals as needed. Use a comma to separate answers as needed.) D. The function is never increasing or decreasing.
Solution
Step 1 ：Given the function \(f(x) = -\frac{1}{x-7}\)
Step 2 ：Find the derivative of the function to determine where the function is increasing or decreasing. The derivative of the function is \(f'(x) = (x - 7)^{-2}\)
Step 3 ：The derivative of the function is always positive except at x=7 where it is undefined. This means the function is increasing for all x not equal to 7.
Step 4 ：The function is never decreasing as there are no values of x for which the derivative is negative.
Step 5 ：\(\boxed{\text{The function is increasing on } (-\infty, 7) \cup (7, \infty). \text{ The function is never decreasing.}}\)
