Problem 75 - Calculus of a Single Variable

Question

Finding Discontinuities In Exercises \(73-76\) , use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ g(x)=\left\\{\begin{array}{ll}{x^{2}-3 x,} & {x>4} \\ {2 x-5,} & {x \leq 4}\end{array}\right. $$

Step-by-Step Solution

Answer

By evaluating the function for \(x=4\), the partition point, we find that the function values from both pieces are different, causing a discontinuity at \(x=4\).

Step 1: Plotting the Function

Using a graphing tool, plot the provided function \(g(x)\). The function is split into two segments: \(x^{2}-3x\) for \(x>4\), and \(2x-5\) for \(x≤4\). We plot these as two separate lines, making sure to mark \(x=4\) as the split point.

Step 2: Analyzing the Graph

After plotting, we can visually check for discontinuities. The point of concern here is at \(x=4\) where the definition of the function changes. Look closely at that point - is the function's value from the left side the same as the value from the right side at \(x=4\)? If they are, the function is continuous; otherwise, it's discontinuous.

Step 3: Calculate the Function Values for \(x=4\)

In order to precisely verify the continuity, we need to calculate \(g(4)\) by using the suitable piece of the function for each side of \(x=4\). For \(x≤4\), use the second equation to compute \(g(4)\), obtaining the left hand limit. Using the first equation for \(x>4\), evaluate, even though \(4>4\) is not true, we call it the right hand limit, which should be compared with the actual function value.

Key Concepts

Piecewise Functions

Piecewise functions are fascinating aspects of mathematical functions. They are defined by different expressions based on their input values. In our specific exercise, we have a piecewise function defined as \( g(x) = \begin{cases} x^{2} - 3x, & x > 4, \ 2x - 5, & x \leq 4. \end{cases} \)
These types of functions are notated in a way that shows different formulas applied over specific ranges of \(x\). Understanding how to approach calculations with piecewise functions is essential because each segment behaves according to its rule. When evaluating or graphing a piecewise function, always check which segment of the function applies to a particular value of \(x\).
Generally, piecewise functions are used to model real-world situations where a rule changes after reaching a certain threshold. They require careful consideration at the points where the function rule changes, often leading us directly to the study of discontinuities.

Graphing Discontinuities

When we graph a piecewise function, we need to pay particular attention to the points where the function's formula switches from one segment to another. These points, known as potential discontinuities, can be tricky. In our example, that critical point is \(x=4\).
To graph the function neatly:

• First, graph the segment \(x^2 - 3x\) for \(x > 4\). This line should not include the point at \(x=4\).
• Next, graph the segment \(2x - 5\) for \(x \leq 4\). Here, ensure that the line includes the endpoint where \(x=4\).

It's essential to pay attention to the boundary behavior at these switching points. Discontinuity appears if there is a break or jump in the graph at these locations. Always check the graph to determine if there is an immediate break or step between differing pieces of the function.
Examining the graph visually gives vital clues - if the distinct segments do not meet or match exactly at the switching point, this indicates a discontinuity.

Continuity at a Point

A function is continuous at a point if there is no interruption in the graph at that location. At \(x=4\), for our piecewise function, we need to determine if the function changes smoothly from one piece to the other.
The continuity at a point can be checked using three conditions:

• The function must be defined at the point of interest (\(x=4\) in this case).
• The limit of the function as \(x\) approaches the point from the left must exist.
• The limit of the function as \(x\) approaches the point from the right must match the function's value from the left and be the same as the value at the point itself.

For the function \( g(x) \), we computed:

• The left-hand limit as \(x\) approaches 4 using \(2x - 5\), which equals 3.
• For \(x > 4\), we don't use \(x=4\) in the calculation, but anticipate the potential value if it were calculated, using \(x^2 - 3x\). However, \(x=4\) strictly applies only to \(2x-5\) as per the piecewise definition.

The results indicate a jump at \(x=4\), marking this as a point of discontinuity, where the different function pieces do not seamlessly connect. Understanding these principles helps in recognizing where discontinuities exist and why certain points are not continuous in piecewise functions.