Problem 28 - College Algebra

Question

Find the vertical asymptotes, if any, of the graph of each rational function. $$r(x)=\frac{x}{x^{2}+3}$$

Step-by-Step Solution

Answer

The function \(r(x) =\frac{x} {x^2+3}\) does not have any vertical asymptotes

Step 1: Identify the Denominator

In this problem, the denominator of the rational function is \(x^2+3\).

Step 2: Set the Denominator Equal to Zero

To find vertical asymptotes, the denominator of the function is set to be equal to zero, this would give us the equation \(x^2+3=0\).

Step 3: Solve for x

Try to solve the equation \(x^2+3=0\) for x.

Step 4: Interpret the Result

The solution of the equation \(x^2+3=0\) does not yield any real roots. Therefore, the function \(r(x) =\frac{x} {x^2+3}\) does not have any vertical asymptotes

Key Concepts

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. In simpler terms, they look like fractions, where the numerator and the denominator are both polynomials. The general form is:

• Numerator: A polynomial on the top.
• Denominator: A polynomial at the bottom.

For example, in the function \(r(x) = \frac{x}{x^2 + 3}\), \(x\) is the numerator while \(x^2 + 3\) is the denominator. Rational functions are known for having unique characteristics like asymptotes and discontinuities, which can be explored further by analyzing the polynomials in both parts.

Denominator

The denominator plays a crucial role in determining the behavior of rational functions. In rational functions, the denominator cannot be zero because dividing by zero is undefined.
When analyzing rational functions, setting the denominator to zero helps identify potential vertical asymptotes.
In \(r(x) = \frac{x}{x^2 + 3}\), the denominator is \(x^2 + 3\). We investigate zeros in rational function denominators because they contribute to vertical asymptotes, points where the graph may go to infinity or negative infinity, depending on the values calculated. Nevertheless, not all functions or unknowns will lead to real results; some may yield complex or no solutions, indicating no vertical asymptote.

Asymptote Determination

Determining asymptotes helps understand the behavior of rational functions as they approach certain values, especially vertical lines known as vertical asymptotes. To find vertical asymptotes analyze where the denominator equals zero:

• Formulate an equation by setting the denominator to zero. In this case, it is \(x^2 + 3 = 0\).
• Solve the equation to find the values of \(x\) where the function becomes undefined, searching for valid real roots.

Vertical asymptotes are present when these solutions have real numbers as the result. If the equation has no real solutions, as found with the equation \(x^2+3=0\), then there are no vertical asymptotes.

Real Roots

Real roots are solutions to equations where the result is a real number, as opposed to an imaginary or complex number.
These roots are significant in determining vertical asymptotes in rational functions.
When solving equations derived from setting the denominator of a rational function to zero, obtaining real roots indicates where vertical asymptotes exist.In the example of \(x^2 + 3 = 0\), solving for \(x\) reveals no real roots because the solution involves taking the square root of a negative number, which results in imaginary numbers such as \(x = i \sqrt{3}\). Hence, the function \(r(x) = \frac{x}{x^2 + 3}\) has no vertical asymptotes, since vertical asymptotes necessitate real roots.