Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the main function's domain. For a piecewise function to be continuous at a point where the sub-functions meet, the left-hand limit and the right-hand limit must both exist and be equal at that point. Additionally, the value of the function at that point must equal the limit. In simpler terms, there should be no breaks or jumps in the graph of the function.
To check for continuity, we primarily focus on the points where the function definition changes, often called 'breakpoints'. We need to ensure that the function values 'match up' at these breakpoints, meaning that the value each piece approaches from its respective side is the same. We also need to verify that the function is actually defined at these breakpoints.
Match the term with its correct definition.
| Term | Definition |
|---|---|
| 1. Piecewise Function | A. The value a function approaches as the input approaches some value. |
| 2. Continuity | B. A function defined by multiple sub-functions on different intervals. |
| 3. Limit | C. The property of a function having no breaks or jumps in its graph. |
| 4. Breakpoint | D. A value within the domain of a function. |
| 5. Domain | E. The point where the definition of a piecewise function changes. |
A piecewise function is said to be _________ at a point if the limit from the left and the limit from the right are _________ and equal to the _________ of the function at that point. The points where the definition of a piecewise function changes are called _________. To check continuity, we must examine the function's behavior at these __________.
Consider a piecewise function defined as follows:
$f(x) = \begin{cases} x^2, & x < 1 \\ ax + b, & 1 \le x < 3 \\ 4, & x \ge 3 \end{cases}$
What values of $a$ and $b$ would make this function continuous everywhere? Explain your reasoning.
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀