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Library of Functions-Target C: Graph a Piecewise-Defined Function

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Graphing a Linear Function with a Restricted Domain (Single Inequality) Graphing a Linear Function with a Restricted Domain (Compound Inequality)

A function is described as a piecewise-defined function because it consists of different pieces. An example of a piecewise function looks like this:

math f(x)= \begin{cases} \text{-}2x-1, \ \ &\mbox{if} \ \ x<-3 \\ 4\text, \ \ &\mbox{if} \ \ \text{-}3\leq x<1 \\ x+4 \ \ &\mbox{if} \ \ x\geq 1 \end{cases}\ math

As you can see, the function consists of 3 pieces: f(x) = -2x - 1, f(x) = 4, and f(x) = x + 4. Each of the three pieces has a specific domain for that part of the function. Here are the domains for each of the three pieces:
 * f(x) = -2x -1: this part of the function has the restricted domain of x < -3
 * f(x) = 4: this piece of the function has the restricted domain of -3 __<__ x < 1
 * f(x) = x + 4: this part of the function has the restricted domain of x __>__ 1

So what does it mean for the function's pieces to have restricted domains? It means that the graph of the function is NOT going to consist of three lines for each of the three pieces of the function. When there are restricted domains within a piecewise-defined function, the graph consists of the following: segments, rays, and points. The following gives some guidelines on how to know what each piece of the graph would look like.


 * If the domain is a single inequality and looks something like this: x __>__ 1, then this portion of the graph will be a ray (with the ray's starting point located at the numeric value in the inequality, i.e. the ray would begin where x equals 1)
 * If the domain is a compound inequality and looks something like this: -2 __<__ x < 3, then this portion of the graph will be a segment (with endpoints located at the two numeric values in the compound inequality, i.e. the endpoints of the segment would be located at -2 and 3)
 * If the domain is an equation and looks something like this: x = 5, then this portion of the graph would be a point (with the point having an x-coordinate of the numeric value in the equation, i.e. the x-coordinate of the point would be 5)

**To learn how to graph linear functions with restricted domains click on the links above** Going back to the example above, when we graph the piecewise defined function, the graph will consist of a ray (x < -3), a segment (-3 __<__ x < 1), and then another ray (x __>__ 1), based off of the different domains of each piece of the function. This is what each of the three pieces would look like individually:

To graph the piecewise function as a whole, take each of the three individual "pieces" and graph them together on the same coordinate plane.



Let's try one more example. Graph the following piecewise-defined function:

math f(x)=\begin{cases} x+3 \ \ &\mbox{if} \ \ x < 1 \\ 2 \ \ &\mbox{if} \ \ x=1 \\ 2x-5 \ \ &\mbox{if} \ \ x > 1 \end{cases}\ math

Look at the different domains for each piece of this function. Since the domains consist of an inequality (x < 1), an equation (x = 1), and then another inequality (x > 1), the graph will consist of a ray, a point, and another ray. This is what each of the three pieces would look like individually: Graphing all three pieces together, will give you the graph of the overall piecewise-defined function:

**Library of Functions-Target C-Piecing Things Together-Quick Check:** Determine what each piece of the following function would like & then graph the function. math f(x)= \begin{cases} \text{-} \dfrac{1}{3}x+2 \ \ &\mbox{if} \ \ x < 0 \\ 3 \ \ &\mbox{if} \ \ 0\leq x<3 \\ \text{-}2x+7 \ \ &\mbox{if} \ \ x\geq 3 \end{cases}\ math

Library of Functions-Target C-Piecing Things Together-Quick Check Solution