Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{2}{2 x-1}\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{2}{2x-1}\) is continuous for all values of x except when \(x = \frac{1}{2}\), which is the point of discontinuity where the denominator equals zero.

Step 1: Identify the function type

First, we should recognize that the given function, \(f(x) = \frac{2}{2x - 1}\), is a rational function. This is because it has the form of \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are both polynomial functions.

Step 2: Examine the denominator

To find the points of continuity, we need to look at the denominator of the function and identify when it equals zero. The denominator of our function is \(2x - 1\). So, let's find when this equals to zero. Set the denominator equal to zero and solve for x: $$2x - 1 = 0$$

Step 3: Solve for x

Add 1 to both sides of the equation and then divide by 2 to solve for x: $$2x = 1$$ $$x = \frac{1}{2}$$

Step 4: Identify the points of discontinuity

We discovered that \(x = \frac{1}{2}\) is the value that makes the denominator equal zero, so the function will be discontinuous at this point.

Step 5: Determine the points of continuity

The function \(f(x) = \frac{2}{2x-1}\) will be continuous for all x ≠ \(\frac{1}{2}\) because rational functions are continuous wherever their denominator is not equal to zero. So, the function is continuous for all values of x except when \(x = \frac{1}{2}\).

Key Concepts

Rational Functions

Rational functions, like the one in our example, are fractions where both the numerator and the denominator are polynomials. The general form of a rational function is expressed as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) is the numerator polynomial and \( Q(x) \) is the denominator polynomial.

These functions can model a wide range of real-world phenomena, including rates and proportional relationships. When graphing rational functions, the shape of the graph can include hyperbolas, vertical asymptotes, horizontal asymptotes, and slant asymptotes depending on the degrees of the polynomials involved. To fully understand these functions, it's essential to analyze and simplify them, ensuring any common factors are canceled out when possible for a clearer view of their behavior across the different domains of x.

Points of Discontinuity

Points of discontinuity are specific x-values where a function does not have a finite or well-defined value. In the context of rational functions, these points are often due to zeros in the denominator, since division by zero is undefined. To find points of discontinuity, we examine the denominator of the function and look for values that would make it zero.

In our example, solving the equation \( 2x - 1 = 0 \) gives us \( x = \frac{1}{2} \), which is the point of discontinuity for \( f(x) \). At this x-value, the function is undefined, leading to a break or gap in the graph. It's important for students to understand how to identify and interpret these points, as they reveal crucial information about the behavior of the function and can have significant implications in calculus for limits and integrals.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the values for the variable that make the equation true. In the context of finding points of discontinuity for a rational function, we set the denominator equal to zero and solve the resulting equation. This process may require various techniques, such as factoring, using the quadratic formula, or employing operations like addition, subtraction, multiplication, and division.

In the given problem, the equation to solve was a simple linear equation \( 2x - 1 = 0 \), leading to the solution \( x = \frac{1}{2} \). A key educational point is that the solutions to the equation are precisely the x-values that we cannot use in the original function because they will make the denominator zero, causing the function to be discontinuous at those points.