Problem 60 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine all values of \(x\) at which the function is discontinuous. \(f(x)=\frac{x^{2}-3 x+2}{x^{2}-2 x}\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x^2 - 3x + 2}{x^2 - 2x}\) is discontinuous at the values \(x = 0\) and \(x = 2\).

Step 1: Write down the function and identify the denominator

We are given the function \(f(x)=\frac{x^{2}-3x+2}{x^{2}-2 x}\). The denominator of this rational function is \(x^2 - 2x\).

Step 2: Solve the denominator equation for x

To find the values of \(x\) that make the function discontinuous, we need to set the denominator equal to zero and solve for \(x\): \[ x^2 - 2x = 0 \]

Step 3: Factor the quadratic expression

Factor the quadratic equation \(x^2 - 2x = 0\): \[ x(x - 2) = 0 \]

Step 4: Solve for x

Now that we have factored the equation, we can solve for the two possible values of \(x\): \[ x = 0 \quad \text{and} \quad x - 2 = 0 \] From the second equation, we find that \(x = 2\).

Step 5: Write down the answer

The function \(f(x) = \frac{x^2 - 3x + 2}{x^2 - 2x}\) is discontinuous at the values \(x = 0\) and \(x = 2\).

Key Concepts

Rational Functions

Rational functions are mathematical expressions involving the division of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
What makes these functions interesting is that they can be complex, but they are very important in mathematics, particularly in calculus. They can describe various real-world phenomena by comparing quantities that change at different rates. For example, the speed of a car (a ratio of distance over time) can be modeled by a rational function.
One key characteristic of rational functions is that they are potentially discontinuous. Discontinuities usually occur where the denominator equals zero because division by zero is undefined. Understanding where these discontinuities exist can help us understand the behavior of the function on its domain.

Denominator Solutions

Finding the solutions to the denominator of a rational function is an essential step for determining discontinuities. This process involves setting the denominator equal to zero and solving for \( x \).
For example, consider the function given in the exercise, where the denominator is \( x^2 - 2x \). We set this equal to zero:

• \( x^2 - 2x = 0 \)

Once we have this equation, our task is to find the values of \( x \) that make **the whole denominator zero**. These solutions point to the x-values where the function becomes undefined, hence the points of discontinuity. Finding where these occur involves techniques like factoring, especially for polynomial denominators.

Factoring Quadratics

Factoring quadratics is a critical skill in solving quadratic equations, simplifying expressions, and finding points of discontinuity in rational functions. A quadratic equation is typically in the form \( ax^2 + bx + c \).
To factor it, we look for two numbers that multiply to the constant term \( c \) and add up to the coefficient \( b \) of the linear term. For instance, in the equation \( x^2 - 2x = 0 \), notice there isn't a constant term. Here, you can factor by pulling out the greatest common factor:

• \( x(x - 2) = 0 \)

The factors tell us that \( x = 0 \) or \( x - 2 = 0 \), leading to solutions \( x = 0 \) and \( x = 2 \).
Factoring helps break down complex expressions and provides insights into the roots of the equation, which are pivotal in determining where functions become discontinuous.