Problem 58 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine all values of \(x\) at which the function is discontinuous. \(f(x)=\frac{1}{(x-1)(x-2)}\)

Step-by-Step Solution

Answer

The function \(f(x) =\frac{1}{(x-1)(x-2)}\) is discontinuous at \(x=1\) and \(x=2\).

Step 1: Identify the Denominator

The denominator of the given function is \((x-1)(x-2)\).

Step 2: Set the Denominator to Zero

In order to find the points of discontinuity, set the denominator equal to zero: \((x-1)(x-2) = 0\)

Step 3: Solve for x

To solve for \(x\), we need to find the values of \(x\) that satisfy the equation \((x-1)(x-2) = 0\). This equation is true if either of the factors is equal to zero. Case 1: \(x - 1 = 0\) Solving for \(x\) gives us the value \(x = 1\). Case 2: \(x - 2 = 0\) Solving for \(x\) gives us the value \(x = 2\).

Step 4: Identify the Points of Discontinuity

The function \(f(x) =\frac{1}{(x-1)(x-2)}\) is discontinuous at \(x = 1\) and \(x = 2\), because at these points the denominator of the function is equal to zero.

Key Concepts

Rational Functions

Rational functions are algebraic expressions that represent the division of two polynomials. The general form of a rational function is expressed as \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \eq 0\). These functions are widely studied in algebra due to their interesting properties, particularly their points of discontinuity.

Rational functions can show different behaviors, such as vertical asymptotes, horizontal asymptotes, and holes, depending on the relationship between the numerator and the denominator. The values of \(x\) that make the denominator \(Q(x)\) equal to zero are the points of discontinuity and usually indicate vertical asymptotes or holes, which are gaps in the graph of the function.

Solving Equations

Solving equations is a fundamental skill in algebra that allows us to find the unknown value of variables that satisfy the given equation. When dealing with rational functions, solving equations often involves identifying the values that cause the denominator to become zero. These calculations are critical for understanding the behavior of the function across its domain.

To solve for the discontinuities in a rational function, we set the denominator equal to zero and solve the resulting equation, as seen in the exercise where the solutions \(x = 1\) and \(x = 2\) are found by solving \( (x-1)(x-2) = 0\). Knowing how to navigate and manipulate equations is essential for analyzing and graphing algebraic functions effectively.

Function Discontinuity

Function discontinuity refers to points at which a function is not continuous. Discontinuities in rational functions often occur where the function is undefined, which is typically at values of \(x\) that result in a zero denominator. These points can indicate different types of discontinuities such as removable discontinuities (holes) or non-removable discontinuities (asymptotes).

In the context of our exercise, the function has discontinuities at \(x = 1\) and \(x = 2\), where the denominator of the function \( \frac{1}{(x-1)(x-2)}\) equals zero. Understanding the nature and classification of these discontinuities helps students graph rational functions accurately and comprehend their limits and behaviors.

Algebraic Functions

Algebraic functions include any function that can be constructed using algebraic operations: addition, subtraction, multiplication, division, raising to powers, and taking roots. These functions can be complex, with multiple operations and variables involved. Rational functions, like the one in our exercise, are a special type of algebraic function where the operation of division leads to interesting properties.

Algebraic functions are broadly explored in mathematics for their diverse applications and the rich insights they provide into mathematical relationships. They help students develop a deep understanding of how quantities relate to each other and how various mathematical concepts are interconnected. Recognizing how discontinuities affect algebraic functions is critical in further studies, such as calculus, as well.