Final answer:
A piecewise function is one composed of multiple sub-functions, each applied over a certain interval. To analyze or graph it, consider each interval and its corresponding function separately, and then combine these on the coordinate plane, ensuring to represent end-points accurately.
Step-by-step explanation:
A piecewise function is a function composed of multiple sub-functions, each of which applies to a certain interval of the function's domain (the set of all possible input values). To work with piecewise functions, you typically handle each case or 'piece' separately and then combine them into one function that is defined for all values of the input.
Example of a Piecewise Function
Consider the function f(x) defined as follows:
2x + 3, if x < 1
x^2, if x ≥ 1
For this piecewise function, you would use the first expression 2x + 3 whenever you have values of x that are less than 1. For values of x that are 1 or greater, you would use the second expression x^2.
Steps to Analyze or Graph a Piecewise Function
- Determine the interval for each piece of the function.
- Analyze the equation that corresponds to each interval.
- Plot the appropriate points and lines or curves for each interval on the same coordinate plane.
- Ensure that the endpoints are accurately represented, considering whether they are included (a solid dot) or excluded (an open dot) from the function.
By following these steps, you can graph or analyze piecewise functions and understand the different behaviors of the function over various intervals.