Problem 46 - Algebra 2

Question

What are the points of discontinuity for \(y=\frac{(2 x+3)(x-5)}{(x+5)(2 x-1)} ?\) $$ \begin{array}{llll}{\text { A. }-5,1} & {\text { B. }-\frac{3}{2}, 5} & {\text { C. }-5, \frac{1}{2}} & {\text { D. } 5,-\frac{1}{2}}\end{array} $$

Step-by-Step Solution

Answer

The points of discontinuity for the function are \( x = -5 \) and \( x = \frac{1}{2} \). Therefore, the correct choice is C. -5, \frac{1}{2}.

Step 1: Identify the denominator of the function

The denominator of the given function is \( (x+5)(2x -1) \).

Step 2: Set the denominator equal to zero

To find the points of discontinuity, we solve the equation \( (x+5)(2x -1) = 0 \). This gives us the values of x for which the function is not defined.

Step 3: Solve the resulting equations

Solving the equations \( x+5 = 0 \) and \( 2x -1 = 0 \) we get \( x = -5 \) and \( x = \frac{1}{2} \), respectively. These are the points of discontinuity for the function.

Key Concepts

Points of Discontinuity

In the context of rational functions, points of discontinuity are values of the variable, usually denoted as \( x \), where the function is not continuous. At these points, the graph of the function has breaks or holes. Such discontinuities occur when the denominator of the rational function equals zero, causing the function to be undefined.
For example, consider the rational function \( y = \frac{(2x+3)(x-5)}{(x+5)(2x-1)} \). The points of discontinuity are found by setting the denominator \((x+5)(2x-1) = 0\) and solving for \( x \). This equation shows that the function is undefined at \( x = -5 \) and \( x = \frac{1}{2} \). If you imagine graphing this function, these points would appear as gaps or breaks where the graph cannot be drawn without lifting your pencil.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomial functions, typically written as \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. Because they're based on polynomial division, the behavior of rational functions is heavily influenced by the solutions to the equation \( q(x) = 0 \).
Here are some key characteristics of rational functions:

• Their domains exclude points where the denominator is zero, leading to discontinuities.
• The degree of the polynomials involved can affect their behavior at infinity or near the asymptotes.
• They often have vertical asymptotes, horizontal asymptotes, or oblique asymptotes depending on the degrees of the numerator and denominator polynomials.

These functions are essential in various fields such as calculus, physics, and engineering due to their ability to model real-world phenomena involving rates or ratios.

Function Undefined Points

For any function, the undefined points are specific values that either make the mathematical operation impossible or lead to indeterminate forms. In rational functions, these points occur when the denominator equals zero. When dealing with rational functions, identifying undefined points is crucial for understanding the function's domain, graph, and behavior.
Using the example \( y = \frac{(2x+3)(x-5)}{(x+5)(2x-1)} \), we determine the function's undefined points by evaluating when \( (x+5)(2x-1) = 0 \), resulting in \( x = -5 \) and \( x = \frac{1}{2} \). At these values, the expression becomes \( \frac{(2x+3)(x-5)}{0} \), which is undefined because division by zero is not possible.
Recognizing these points is important as they can affect the continuity of the function and are excluded from the domain. They also help in visualizing potential asymptotes or discontinuities on the graph.