Problem 56 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The function \(f(x) = \frac{|x-1|}{x-1}\) is continuous for \(x < 1\) and \(x > 1\), but discontinuous at \(x = 1\).

Step 1: Analyze the absolute value function

The function's numerator is \(|x-1|\), which is the absolute value function. Recall that an absolute value function is continuous everywhere, so we only need to focus on its denominator.

Step 2: Investigate the denominator

The denominator is \(x-1\). We need to find cases when the denominator would be zero or undefined, which would make the whole function discontinuous. \(x-1 = 0\) when \(x = 1\). So, when \(x=1\), the function will be undefined because of division by zero. Thus, this is the point of discontinuity.

Step 3: Determine the continuous regions of the function

Now that we have established the point of discontinuity, let's find the continuous regions of the function: 1. When \(x < 1\): In this interval, the function \(f(x) = \frac{|x-1|}{x-1}\) can be rewritten as \(f(x) = \frac{(1-x)}{x-1} = -1\). The function \(f(x) = -1\) is continuous for this interval. 2. When \(x > 1\): In this interval, the function \(f(x) = \frac{|x-1|}{x-1}\) can be rewritten as \(f(x) = \frac{(x-1)}{x-1} = 1\). The function \(f(x) = 1\) is continuous for this interval. Thus, the function is continuous for \(x < 1\) and \(x > 1\). The only discontinuous point of the function is at \(x = 1\).

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, is a core concept in mathematics that encapsulates the distance of a number from zero on the number line, regardless of direction. Essentially, it converts any negative number into its positive counterpart, while positive numbers and zero remain unchanged.

Take the function in our exercise: the numerator is given by the absolute value of the expression |x - 1|. The graph of an absolute value function is typically V-shaped, opening upwards. This particular function mirrors across the line x = 1. An important characteristic of absolute value functions is its continuity; they do not have breaks or holes, hence they are continuous for all real numbers.

However, the continuity of a function involving absolute values can become more complex when combined with other expressions, as seen in our example of \(f(x) = \frac{|x-1|}{x-1}\). Here, the denominator dictates where the function might face issues of discontinuity.

Point of Discontinuity

A point of discontinuity occurs where a function is not continuous. Discontinuities arise where functions break, jump, or have asymptotes — essentially any point where the function isn't well-behaved and can't be smoothly drawn. For rational functions, this often happens where there is a division by zero.

In our exercise, we carefully examine the denominator, x - 1. This expression equals zero precisely when x is 1, causing the function to be undefined at that point because of division by zero. Therefore, x = 1 is our point of discontinuity for the function \(f(x) = \frac{|x-1|}{x-1}\).

Identifying discontinuities is crucial because it tells us where the function cannot output a value. It is a signal that something notable is occurring at that specific input, requiring us to pay special attention or potentially adjust our function to manage or understand this behavior.

Continuous Regions

Continuous regions of a function are intervals where the function is defined and has no breaks, jumps, or holes, implying that you can draw the graph without lifting your pencil. For the function in our exercise, continuous regions are any values of x except for the point of discontinuity.

To be more technical, we split our analysis into two cases based on the value of x relative to our point of discontinuity, x=1. When x is less than 1, the function simplifies to a constant function f(x) = -1; when x is greater than 1, the function simplifies to another constant function, f(x) = 1. Both constant functions are continuous across their domains, which in these cases exclude the single point x=1.

This gives us two continuous regions: x < 1 and x > 1. These are intervals where our function behaves nicely and where there are no interruptions in its graph. Understanding continuous regions is essential for sketching accurate graphs and for deeper comprehension of a function's behavior across its domain.