### Home > APCALC > Chapter 1 > Lesson 1.3.2 > Problem1-117

1-117.

For each part below, give an example of a function with specified attributes. Provide a sketch of each function.

1. A function with a hole at $x = 3$ and an asymptote at $x = −1$.

Write a rational function with $(x − 3)$ in both the numerator and the denominator to make a hole at $x = 3$.

Remember that vertical asymptotes are formed by zeros in the denominator.

$\frac{x-3}{(x-3)(x+1)}$

2. A function with asymptotes at the $y$-axis and $x = 5$ and a hole at $x = −4$.

An asymptote at the $x$-axis is the same as having a vertical asymptote of $x = 0$.

3. A function with an end-behavior function $g(x) = 3x − 1$.

Holes and vertical asymptotes do not affect end behavior. How can $y = 3x − 1$ be altered so that it has a hole or vertical asymptote?

Possible answer: $y = 3x − 1 + \text{[answer to part (a)]}$.