Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(g(x)=\frac{x^{2}-9 x+20}{x^{2}-16}\)

Step-by-Step Solution

Answer

The function \(g(x)=\frac{x^{2}-9 x+20}{x^{2}-16}\) is continuous on the intervals \((- \infty, -4) \cup (-4, 4) \cup (4, \infty)\). It is not continuous at \(x = -4\) and \(x = 4\) because these values make the denominator of the function equal to zero, causing the function to be undefined at those points.

Step 1: Identify the denominator

The denominator of the rational function is \(x^{2}-16\). We need to solve for \(x\) when the denominator is equal to zero.

Step 2: Solve the equation

We set the equation \(x^{2}-16 = 0\). The solutions are \(x = -4\) and \(x = 4\), found by factorization or by using the quadratic formula.

Step 3: Determine the intervals of continuity

Since the function is not defined at \(x = -4\) and \(x = 4\), we say that the function is continuous on the intervals \((- \infty, -4) \cup (-4, 4) \cup (4, \infty)\).

Key Concepts

Rational Functions

Rational functions are a type of function that can be written as a ratio of two polynomials. Specifically, a rational function has the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. The numerator, \( P(x) \), can be a zero polynomial which simplifies the expression, while the denominator, \( Q(x) \), must not be zero.

Rational functions are important as they model many natural phenomena and are fundamental in calculus and algebra. Understanding them involves studying where the denominator equals zero, as these are points of discontinuity.

Not every rational function is defined for all real numbers. Whenever the denominator \( Q(x) \) equals zero, the function becomes undefined. It's essential to solve for these values to fully understand the behavior of the function on the real line. Let's explore further how the real line concept relates to these functions.

Real Line

The real line is a concept used to visually represent real numbers in a continuum. Imagine a straight line that extends endlessly in both directions. Each point on this line corresponds to a real number.

When considering the continuity of a function like our given rational function \( g(x)=\frac{x^{2}-9x+20}{x^{2}-16} \), we examine points along the real line where \( g(x) \) is defined. Examining the real line helps us understand the intervals over which the function is continuous.

For any given rational function, certain points may cause the denominator to become zero, leading to discontinuities. By identifying such values and omitting them, we can describe where the function is continuous. For example, in our function, by excluding \( x = -4 \) and \( x = 4 \) from the real line, we determine the intervals \((- \infty, -4) \cup (-4, 4) \cup (4, \infty)\) where the function is continuous.

Discontinuity

Discontinuity in a function occurs at points where the function is not smoothly connected or becomes undefined. In the case of rational functions, discontinuities typically happen where the denominator is equal to zero.

Identifying discontinuities is critical for understanding how a function behaves on the real line. Consider our rational function example: \( g(x)=\frac{x^{2}-9x+20}{x^{2}-16} \). Here, the denominator \( x^{2}-16 \) equals zero at \( x = 4 \) and \( x = -4 \).

At these points, the function experiences removable discontinuities because these are the only values that make the denominator zero and thus the function undefined. These discontinuities lead us to divide the real line into intervals where the function remains continuous: \((- \infty, -4) \cup (-4, 4) \cup (4, \infty)\).

Understanding and finding these discontinuities help us predict the behavior of rational functions and solve problems related to their continuity.