Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Explain why the graph of the rational function \(f(x)=\frac{-1}{x^{2}+4}\) has no vertical asymptotes.

Step-by-Step Solution

Answer

The function has no vertical asymptotes because the denominator, \(x^2 + 4\), never equals zero for real numbers.

Step 1: Identify the Function Type

The given function is a rational function of the form \(f(x) = \frac{g(x)}{h(x)}\), where \(g(x) = -1\) and \(h(x) = x^2 + 4\). Rational functions can have vertical asymptotes where the denominator is zero.

Step 2: Set the Denominator to Zero

To find potential vertical asymptotes, set the denominator \(h(x)\) equal to zero: \(x^2 + 4 = 0\).

Step 3: Solve for x

Attempt to solve \(x^2 + 4 = 0\) for \(x\). Subtract 4 from both sides to get \(x^2 = -4\). Notice that \(x^2 = -4\) has no real solutions because the square of a real number cannot be negative.

Step 4: Conclude on Vertical Asymptotes

Since there are no real solutions to \(x^2 + 4 = 0\), the denominator never becomes zero for real numbers. Therefore, the graph of the function \(f(x)\) has no vertical asymptotes.

Key Concepts

Rational Functions

A rational function is simply a ratio of two polynomials. You can think of it like a fraction where both the numerator and the denominator are polynomials. In the function given here, \(f(x) = \frac{-1}{x^2 + 4}\), the numerator \(-1\) is a constant polynomial, and the denominator \(x^2 + 4\) is a quadratic polynomial.

Rational functions are common in algebra, and they have some interesting properties, especially when it comes to graphs. One important feature they often have is **vertical asymptotes**, which occur where the function becomes undefined. To identify potential vertical asymptotes, we focus on the denominator because it's where the "trouble" is! Let's explore this further.

Denominator

The denominator in a rational function plays a key role in determining its properties, particularly the vertical asymptotes. In our function, the denominator is \(x^2 + 4\). Rational functions have vertical asymptotes at the values of \(x\) that make the denominator zero.

Let's work through this:

• Set \(h(x) = x^2 + 4\) equal to zero to find potential asymptotes.
• Solve the equation \(x^2 + 4 = 0\).

Now, as you try to solve this, you will notice that subtracting 4 from both sides gives \(x^2 = -4\). But hang on, a key mathematical concept here is that the square of a real number can't be negative. No real number when squared will yield \(-4\), hence \(x^2 + 4 = 0\) has no real solutions. This means the denominator never becomes zero for any real number, leading to no vertical asymptotes.

Real Solutions

To determine vertical asymptotes, finding real solutions to the equation \(x^2 + 4 = 0\) was critical. But as we explored earlier, this equation has no real solutions.Here's why:

• Real solutions are values of \(x\) that satisfy an equation using real numbers.
• In our context, \(x^2 + 4 = 0\) simplifies to \(x^2 = -4\), which isn't possible using real numbers since squares can't be negative.

This absence of real solutions means there is no value of \(x\) making the denominator zero on the real number line. Consequently, \(f(x)\) does not have vertical asymptotes. Understanding this will help you graph rational functions accurately and recognize why certain functions behave in unique ways on their graphs.