Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\frac{x+4}{x^{2}-6 x+5}\)

Step-by-Step Solution

Answer

The function \(f(x) =\frac{x+4}{x^{2}-6 x+5}\) is continuous on the entire real line except at \(x = 1\) and \(x = 5\).

Step 1: Identify the Domain of the Function

The domain of the function is all the real numbers except those values of \(x\) for which the denominator becomes zero. Solve the quadratic equation \(x^{2} - 6x + 5 = 0\) to find these values. This will give you the values of \(x\) for which the function is undefined.

Step 2: Solve the Quadratic Equation

Begin by factoring the quadratic equation: \(x^{2} - 6x + 5 = 0\) becomes \((x-1)(x-5) = 0\). This gives us roots \(x = 1\) and \(x = 5\). Therefore, the function is undefined at \(x = 1\) and \(x = 5\).

Step 3: Examine Continuity on the Domain

The function is a rational function and rational functions are continuous wherever they are defined. Therefore, the given function is continuous on \((-\infty, 1) \cup (1, 5) \cup (5, \infty)\), which is the entire real line with the exception of \(x = 1\) and \(x = 5\).

Key Concepts

Rational Functions

A rational function is a type of function represented by the ratio of two polynomials. The general form of a rational function is \( R(x) = \frac{P(x)}{Q(x)} \), where \(P(x) \) and \(Q(x) \) are polynomials and \(Q(x)\) is not zero.

Rational functions have a few key characteristics:

• The domain of a rational function is all real numbers except where the denominator \(Q(x) \) is zero.
• They can display vertical and horizontal asymptotes depending on the degrees of the polynomials in \(P(x) \) and \(Q(x) \).
• Rational functions are continuous wherever they are defined, meaning there are no breaks, holes, or jumps in the graph except potentially where \(Q(x) \) equals zero.

By understanding these properties, we can look at any rational function, like \(f(x)=\frac{x+4}{x^2-6x+5}\), and determine where it is continuous based on its domain.

Domain of a Function

The domain of a function is the complete set of all possible input values \(x\) that allow the function \(f(x)\) to be defined. For rational functions specifically, the domain is affected by the denominator. The function is not defined where the denominator is zero.

To find the domain of \(f(x)=\frac{x+4}{x^2-6x+5}\), follow these steps:

• Solve \(x^2-6x+5=0\) to find the values of \(x\) that make the denominator zero.
• The solutions \(x = 1\) and \(x = 5\) are excluded from the domain.

This means the domain of \(f(x)\) is all real numbers except \(x = 1\) and \(x = 5\). Thus, the function is continuous everywhere else: \((\text{-}\infty, 1) \cup (1, 5) \cup (5, \infty)\).

Identifying the domain is crucial because it tells us where the function can operate without running into undefined points.

Quadratic Equations

Quadratic equations are second-degree polynomial equations of the form \( ax^2 + bx + c = 0 \). They are standard because they appear in various mathematical scenarios, including finding the roots that make a denominator of a rational function zero.

Solving a quadratic equation can be done by:

• Factoring, which involves expressing it as a product of two binomials.
• Completing the square, a method used to derive the quadratic formula.
• Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), which provides the solutions directly.

For the equation \( x^2 - 6x + 5 = 0 \), factoring yields \((x - 1)(x - 5) = 0\), leading to roots \(x = 1\) and \(x = 5 \). These values are critical since they determine where functions, like the given rational function, are undefined.