Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\left\\{\begin{array}{ll}x^{2}-4, & x \leq 0 \\ 3 x+1, & x>0\end{array}\right.\)

Step-by-Step Solution

Answer

The function \(f(x)\) is continuous on the intervals \( (-\infty, 0) \) and \( (0, \infty) \). The function is discontinuous at \(x = 0\).

Step 1: Understanding the Function's Definition

The function \(f(x)\) is piecewise defined as \(x^{2}-4\) when \(x \leq 0\) and \(3x+1\) when \(x>0\). So, function is divided into two parts. We have to determine the intervals where it is continuous.

Step 2: Investigating Continuity in Intervals

To start with analyzing the intervals of the piecewise function, we should examine each part separately. For \(x \leq 0\), we have the function \(f(x) = x^{2}-4\), which is a quadratic function and hence continuous everywhere. Therefore, \(-\infty < x \leq 0\) is an interval where the function is continuous. For \(x>0\), the function \(f(x) = 3x + 1\) is a linear function, which is also continuous everywhere. So, \(0 < x < \infty\) is the next interval where the function exhibits continuity.

Step 3: Candidate for Discontinuity

The function may exhibit discontinuity where the two pieces of the function meet, that is at \(x = 0\). On plugging \(x = 0\) in both pieces of function, \(f(x) = x^2 -4\) and \(f(x) = 3x + 1\), we get \(-4\) and \(1\) respectively, which are different. For a function to be continuous at a certain point, it must satisfy the three conditions of continuity: it must be defined at that point, the limit must exist at that point, and the function value must equal the limit. Here, since the function values are different and do not equal the limit, the function is discontinuous at \(x = 0\).

Step 4: Conclusion

The function \(f(x)\) is therefore continuous on interval \( (-\infty, 0) \) and \( (0, \infty) \). However, it is discontinuous at \(x = 0\) because it does not satisfy the necessary conditions for continuity.

Key Concepts

Piecewise Functions

A piecewise function is a type of function defined by multiple sub-functions, each applying to a different domain, or "piece" of the overall function's domain. In simple terms, you can think of these as a puzzle where each piece fits in a separate section.

For example, the function given in the original exercise, \[f(x)=\left\{\begin{array}{ll}x^{2}-4, & x \leq 0 \ 3 x+1, & x>0\end{array}\right.\]This tells us that for values of \(x\) less than or equal to zero, one formula is used, and for values greater than zero, another formula takes over.

These types of functions are handy when trying to model situations where a rule changes at different intervals such as taxes, rates of change in a science experiment, and more.

Interestingly, each segment of a piecewise function might have different characteristics, including how they approach continuity.

Discontinuity

Discontinuity in mathematics refers to a point at which a function is not continuous. For functions like the one described in the exercise, you might encounter discontinuity at the point where two pieces of a piecewise function meet.

In our exercise, at \(x = 0\), each segment of the piecewise function, \(x^{2}-4\) and \(3x+1\), does not "meet" in a way that aligns perfectly.
When evaluating these at \(x=0\), we get values of \(-4\) and \(1\) respectively.

• The first condition of continuity fails because the function does not have the same value from both sides at \(x=0\).
• The limit from the left does not equal the limit from the right.

Since these values differ, there's a break or "jump" between them, causing a discontinuity.

Understanding discontinuities is essential. It helps us know where a mathematical model might have limitations or where a real-world application might experience an abrupt change.

Intervals of Continuity

Finding the intervals of continuity is all about determining where a function behaves smoothly without any jumps or breaks.

In our piecewise function example, the segments each have their continuous zones:

• For \(x \leq 0\), \(x^{2}-4\) is a quadratic function, which is continuous across its entire domain. Therefore, it implies that \(-\infty < x \leq 0\) is a continuous interval.
• For \(x>0\), \(3x+1\) is linear and continuous, making \(0 < x < \infty\) the next continuous interval.

By examining these, you can confidently state where the function is continuous and note any exceptions.

Remember that continuity implies no abrupt breaks or changes, and each function or sub-part within a piecewise definition provides clues about where they occur.