Problem 128 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

Step-by-Step Solution

Answer

The statement 'The graph of a rational function can have three vertical asymptotes' is true.

Step 1: Understanding the properties of a rational function

A rational function is any function that can be described as the ratio of two polynomial functions. The vertical asymptotes of a rational function are the x-values that make the denominator of the function equals zero (assuming the numerator is not zero at these points).

Step 2: Analyzing the given statement

The given statement is 'The graph of a rational function can have three vertical asymptotes.' To determine whether this statement is true or false, it is critical to examine the possibility for a rational function to have three distinct x-values which cause the denominator to be zero.

Step 3: Verifying the statement

A rational function can indeed have three vertical asymptotes. For example, consider the function \( f(x) = \frac{1}{(x-a)(x-b)(x-c)} \), where a, b, and c are distinct real numbers. In this case, a, b, and c will be the vertical asymptotes because the function will approach infinity at these points. Therefore, we conclude that the given statement is true.