This article will discuss the graph that represents a piecewise-defined function. The concept of a piecewise-defined function can be hard to understand at first, but with some practice it’s easy to master. This article will explain the basics of a piecewise-defined function, provide examples of such functions, and then discuss how to graph these functions.
A piecewise-defined function is a function that has different formulas that produce different outputs over different pieces of its domain. The pieces of the domain in which the function operates are often referred to as “intervals”, as each interval produces a different output for a given input. Piecewise-defined functions are particularly useful when an equation has a large domain, as it allows us to break the equation into smaller, more manageable chunks.
Let’s look at an example of a piecewise-defined function. Suppose we have the following equation:
f(x) =
-2x, & x < -1 \
-1, & -1 ≤ x < 2 \
x-1, & x ≥ 2
This function has three pieces: one for the interval x < -1, one for -1 ≤ x < 2, and one for x ≥ 2. As you can see, each piece has a different equation associated with it, so the output of the function will depend on which interval x is in.
Now that we understand what a piecewise-defined function is, let’s look at how to graph it. In order to graph a piecewise-defined function, we need to plot the points of each part of the equation. We can do this by finding the x-intercepts from each piece and plotting those points. For example, in the equation above, we have three x-intercepts: -2, -1, and 1. We can plot those points on a graph and then draw a line between them. The result is the graph pictured below:
This graph represents the piecewise-defined function that we discussed earlier.
In conclusion, a piecewise-defined function is a function that has different formulas associated with it over different intervals of its domain. It can be tricky to graph piecewise-defined functions at first, but with a bit of practice, it can become second nature. If you’re ever unsure of how to graph a piecewise-defined function, just remember to plot the points of each piece and then draw a line connecting them.