Problem 47 - Algebra 2

Question

How many points of discontinuity does the function \(y=\frac{2 x}{x^{2}+4}\) have? $$ \begin{array}{lllll}{\text { F. } 0} & {\text { G. } 1} & {\text { H. } 2} & {\text { J. } 4}\end{array} $$

Step-by-Step Solution

Answer

F. 0

Step 1: Set the Denominator to Zero

The first step is to set the denominator of the function, \(x^{2}+4\), equal to zero and solve for \(x\). So we get, \(x^{2}+4 = 0\).

Step 2: Solve the equation

Next, we rearrange the equation and solve for \(x\). Subtracting 4 from both sides we get, \(x^{2} = -4\). When we take square root on both sides, normally we would have two solutions, \(x = \sqrt{-4}\) and \(x = -\sqrt{-4}\). However, the square root of a negative number is not a real number, it is a complex number. Therefore, in the real number system there is no solution.

Step 3: Conclusion of the findings

Since there are no real values of \(x\) that make the denominator zero, it can be concluded that the function has no points of discontinuity in the real number system.

Key Concepts

Rational Functions

Rational functions are essentially the division of two polynomial expressions, where the numerator and the denominator are both polynomials. For the function to be considered rational, the denominator must not be equal to zero, as this would make the function undefined. In the given exercise, the function is expressed as \( y = \frac{2x}{x^2 + 4} \).

• The numerator here is \( 2x \).
• The denominator is \( x^2 + 4 \).

Understanding rational functions involves determining where they are undefined, which often occurs where the denominator is zero. These points are initially suspected as points of discontinuity. For the rational function in the example, the challenge is to find such points by solving \( x^2 + 4 = 0 \), which requires a grasp of complex numbers.

Complex Numbers

Complex numbers arise when dealing with the square roots of negative numbers, which have no solutions in the real number system. They are represented in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit equal to \( \sqrt{-1} \).In the exercise, solving \( x^2 = -4 \) involves taking the square root of a negative number, which implies the function yields complex solutions. The solutions are \( x = \sqrt{-4} = 2i \) and \( x = -2i \).

• This signifies that zeros of the denominator are not on the real number line but on the complex plane.
• Complex numbers are not part of the real number solutions, indicating no real number discontinuities in the function.

Rational functions are typically analyzed within the realm of real numbers, hence no discontinuities are observed here given that the zeros lie in the complex number space.

Real Number System

The real number system consists of all numbers that can be found on the number line, including all rational and irrational numbers like integers, fractions, and radicals. It does not include complex numbers, which exist outside this system.In the context of rational functions, discontinuities are generally considered over the real number system. Solving \( x^2 + 4 = 0 \) within this system provides no solutions, as we cannot have \( x^2 = -4 \) with real numbers. Thus:

• No x-values within the real number framework make the function undefined, implying no real discontinuities.
• The search for solutions within the real number system is crucial prior to considering any potential impacts from complex numbers.

Since the denominator does not equal zero for any real x-values, there are zero points of discontinuity in this system for the provided function.