Problem 5 - College Algebra with Corequisite Support

Question

Can a graph of a rational function have no \(x\) -intercepts? If so, how?

Step-by-Step Solution

Answer

Yes, if the numerator of the rational function is always non-zero.

Step 1: Identify Definition of X-intercepts

The x-intercepts of a graph of a function are the points where the graph crosses the x-axis. For rational functions, these occur where the numerator equals zero and the denominator is not zero.

Step 2: Analyze Conditions for No X-intercepts

For a rational function to have no x-intercepts, its numerator must never be zero. This means there are no values of x that make the numerator zero, or any zero values of the numerator are canceled out by zero values in the denominator.

Step 3: Example of a Rational Function With No X-intercepts

Consider the rational function \( f(x) = \frac{2}{x^2 + 1} \). Here, the numerator is constant (2), and it never equals zero, while the denominator \( x^2 + 1 \) is always positive for all real numbers \( x \). Thus, this function has no x-intercepts.

Key Concepts

x-intercepts

X-intercepts are the points where a graph crosses the x-axis. At these points, the value of the function is zero. For rational functions, x-intercepts occur where the numerator of the function is zero, and the denominator is not zero. Imagine a rational function like \( f(x) = \frac{a(x)}{b(x)} \). Here, x-intercepts are found by solving \( a(x) = 0 \) while ensuring that \( b(x) eq 0 \).
Understanding x-intercepts helps us analyze the behavior of rational functions. In cases where the numerator never equals zero, the graph will not intersect the x-axis. For example, if the numerator is a non-zero constant, like in \( f(x) = \frac{2}{x^2 + 1} \), the function will never meet the x-axis, meaning there are no x-intercepts at all. This particular setup allows students to appreciate how certain combinations of numerators and denominators influence the graph’s intersections with axes.

numerator and denominator

For rational functions, the numerator and denominator are key components in determining the function's behavior. A rational function has the form \( f(x) = \frac{n(x)}{d(x)} \), where \( n(x) \) is the numerator and \( d(x) \) is the denominator. Each part has a distinct role:

• The numerator \( n(x) \) determines the value at which the function's graph could potentially cross the x-axis, provided the denominator is not zero at the same points.
• The denominator \( d(x) \), on the other hand, cannot be zero, as this would make the function undefined. Points where \( d(x) = 0 \) are often points of discontinuity, like holes or vertical asymptotes.

In a function like \( f(x) = \frac{2}{x^2 + 1} \), the numerator is a constant and never zero, and the denominator is always positive. This absence of zeros in the numerator results in no x-intercepts, while the denominator remaining positive prevents vertical asymptotes but ensures the function is continuous over all real numbers. Understanding the interplay between the numerator and denominator helps to predict the graph's behavior and identify key characteristics such as intercepts or discontinuities.

graph of a function

The graph of a rational function illustrates the relationship between its input and output values visually. Key features of a graph of a rational function include:

• Intercepts, where the graph crosses the axes. The y-intercept is found by evaluating the function at \( x = 0 \), while x-intercepts depend on where the numerator is zero.
• Asymptotes, which are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or slant asymptotes describe the behavior as \( x \) goes to positive or negative infinity.
• Holes, which appear when both the numerator and denominator are zero at the same point, indicating removable discontinuities.

The function \( f(x) = \frac{2}{x^2 + 1} \) provides a good example. Its graph does not touch the x-axis, indicating no x-intercepts. There are no vertical asymptotes since \( x^2 + 1 \) is never zero. Instead, the graph is smooth and continuous, encapsulating the function's behavior in a way that equations alone might not fully convey. This visual approach clarifies the function's behavior across different values and provides insights into its analytical properties.