Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(f(x)=\left\\{\begin{array}{ll}3 x-1, & x \leq 1 \\ x+1, & x>1\end{array}\right.\)

Step-by-Step Solution

Answer

The function \(f(x)\) is not continuous at \(x = 1\) because the function is defined by two different expressions at this \(x\) - value and hence, there is a 'jump' or discontinuity at \(x = 1\).

Step 1: Understand the piecewise function

A piecewise function is a function that is defined by several subfunctions, each corresponding to a certain interval of the independent variable. Here, the function \(f(x)\) is defined by two different expressions; \(3x - 1\) for \(x \leq 1\) and \(x + 1\) for \(x > 1\).

Step 2: Plot the piecewise function

First, graph the function \(3x - 1\) on the interval \(x \leq 1\). Then, graph the function \(x + 1\) on the interval \(x > 1\). Remember to include the breakpoint \(x = 1\) in the graph.

Step 3: Analyze the graph for discontinuities

Look for any breaks, jumps, or disconnects along the graph which would indicate points of discontinuity. A function is not continuous at a point if it is not defined at that point, or if a limit does not exist at that point.

Step 4: Identify the \(x\) - value(s) at which the function is not continuous

Determine any \(x\) - value(s) at which the piecewise function is not continuous from the graph. In a piecewise function, potential discontinuity may occur at the breakpoints, i.e., where the definition of the function changes.

Key Concepts

Graphing Piecewise Functions

Graphing piecewise functions might appear challenging at first, but with a systematic approach, it can be simplified. A piecewise function is defined by multiple sub-functions, each applicable over certain segments of the domain.
The function given, \(f(x) = \begin{cases} 3x-1, & x \leq 1 \ x+1, & x>1 \end{cases}\), involves two distinct linear expressions. For \(x \leq 1\), the function is \(3x-1\), and for \(x > 1\), the function is \(x+1\).
Begin graphing by managing each segment separately:

• Start with \(3x-1\) when \(x \leq 1\). Plot by finding key points like \(x=1\) and check its corresponding \(y\)-value, which is 2.
• Next, graph \(x+1\) for \(x > 1\). Select \(x=2\) and place its point at \((2, 3)\).

Use these points and connect them to form linear sections for each part of the function. Ensure that you reflect any changes clearly at the breakpoint \(x=1\). This way, the complete picture of the piecewise function emerges clearly.

Discontinuities in Functions

Discontinuities are specific points where a function behaves unexpectedly. Such points might be where the function isn't defined, or when the graph jumps suddenly. In piecewise functions, discontinuities may arise at the breakpoints where the function changes its defining equation.
Analyze the piecewise function \(f(x)\) at \(x=1\), which is the breakpoint here. Here’s how you can assess it:

• For the left section \(3x-1\), as \(x\) approaches 1, \(f(x) = 2\).
• For the right section \(x+1\), as \(x\) approaches 1, \(f(x) = 2\) also.

Both sections meet smoothly at \(x=1\) with the same function value. This confirms no jump or break at this point. However, remember, had these values differed, it would signal a discontinuity. Identifying such spots helps in understanding the behavior and nature of functions.

Limits and Continuity

Understanding limits is essential for analyzing continuity in functions. When evaluating a limit, we explore the function’s behavior as \(x\) nears a particular point. For piecewise functions, these evaluations determine continuity.
In our example, consider the function \(f(x)\) at the breakpoint \(x=1\):

• Find the limit from the left: As \(x\) approaches 1 from the left (\(x\leq1\)), \(f(x) = 3x - 1\), so the limit is 2.
• Find the limit from the right: As \(x\) approaches 1 from the right (\(x>1\)), \(f(x) = x + 1\), thus the limit is 2.

Since both left-hand and right-hand limits match, \(f(x)\) is continuous at \(x=1\). Continuity happens when a point's limit exists and equals the function's value at that point. This ensures no sudden changes or gaps, allowing the function’s graph to be drawn without lifting your pencil. This is a cornerstone for understanding continuous behavior in various functions.