Problem 80 - Algebra 2

Question

For each rational function, find any points of discontinuity. $$ y=\frac{2 x-1}{x^{2}+4} $$

Step-by-Step Solution

Answer

There are no points of discontinuity for the function \(y=\frac{2x-1}{x^{2}+4}\) since no real x-values would make the denominator equal to zero.

Step 1: Identify the denominator

The denominator of the function \(y=\frac{2x-1}{x^{2}+4}\) is \(x^{2}+4\). It's important to isolate this for inspection.

Step 2: Set the denominator equal to zero

Now it's necessary to solve for x by setting \(x^{2}+4=0\). The solution for this equation would indicate the values of x which would make y undefined.

Step 3: Solve the equation

To solve for x, it's needed to subtract 4 from both sides of the equation: \(x^{2} = -4\). This result is indicative of no real x values since the square root of a negative number is not real. This would suggest there are no x-values which will make y undefined.

Key Concepts

Discontinuity in Rational Functions

In the context of rational functions such as \( y = \frac{2x-1}{x^2+4} \), understanding points of discontinuity is essential. Discontinuity occurs in places where a function cannot be smoothly drawn or graphed. For rational functions, these are typically points where the function is undefined due to division by zero. If the denominator becomes zero at any value of \( x \), this would suggest a point of discontinuity.
However, in the example given, after following the steps to solve \( x^2 + 4 = 0 \), it turns out that \( x^2 = -4 \). Because the square root of a negative number does not yield any real numbers, the function does not have any real x-values that make the denominator zero. Thus, there are no points of discontinuity for real values in this function.
Learning to find points of discontinuity helps in properly understanding the behavior of a graph and knowing where any breaks might occur.

Understanding the Denominator

The denominator of a rational function is crucial because it defines where the function might become undefined. In the function \( y = \frac{2x-1}{x^2+4} \), the denominator is \( x^2+4 \). This expression must be thoroughly examined to determine if there are any values of \( x \) that could zero it out.
- A zero in the denominator results in the function being undefined at that particular value.
In rational functions, the denominator being zero indicates vertical asymptotes or holes, leading to points of discontinuity. Regularly investigating the denominator for any potential zero values is necessary for analyzing a function's domain.
By setting the denominator equal to zero, you can solve for x and find possible discontinuity points. In this case, since the denominator can never equal zero for real numbers, the function maintains its continuity across all real values.

Identifying Undefined Values

Undefined values in rational functions are typically associated with values of \( x \) that make the denominator zero. If a denominator is zero at any particular \( x \), the function itself cannot exist at that point, as division by zero is undefined in mathematics.
For the rational function \( y = \frac{2x-1}{x^2+4} \), solving \( x^2+4=0 \) to find undefined values reveals no real solutions, meaning \( x \) values remain defined across the number line.

• Undefined values arise when the denominator equals zero.
• For the function \( y = \frac{2x-1}{x^2+4} \), this situation does not occur with real x-values.

This highlights the importance of scrutinizing the denominator for potential zeroes, as these directly correlate with where the function might otherwise be undefined. Understanding undefined values is fundamental in plotting rational functions and determining their overall behavior.