Graphing piecewise functions
A piecewise function is a function that is in pieces. You'll notice that although there can be several different pieces, we must be careful not to have several outputs (y values) belonging to a single input (x value).
Look at the piecewise function to the right. Assuming that each box is one unit, it appears that at an x value of 2, there are three different pieces of the function overlapping. But, we know that can't be the case or this wouldn't be a function. This is when closed and open dots come back into play from our domain graphs (see Project 1, page 2). If the dot is open, the coordinate is excluded, if it is closed than it is included. You'll notice that only (2,6) is actually included in the domain.
If we were to write an equation for the graph to the right, it would look like this:
x², x<2
f(x) { 6, x = 2
10-x, 2<x<6
Notice that the domain is determined by the comma and the x notation afterwards. When graphing, try graphing the lines first, then erase and make dots and arrows as dictated by the domain (the x value behind the comma). Always check at the end to ensure that your function is a function.
Look at the piecewise function to the right. Assuming that each box is one unit, it appears that at an x value of 2, there are three different pieces of the function overlapping. But, we know that can't be the case or this wouldn't be a function. This is when closed and open dots come back into play from our domain graphs (see Project 1, page 2). If the dot is open, the coordinate is excluded, if it is closed than it is included. You'll notice that only (2,6) is actually included in the domain.
If we were to write an equation for the graph to the right, it would look like this:
x², x<2
f(x) { 6, x = 2
10-x, 2<x<6
Notice that the domain is determined by the comma and the x notation afterwards. When graphing, try graphing the lines first, then erase and make dots and arrows as dictated by the domain (the x value behind the comma). Always check at the end to ensure that your function is a function.
Let's write an equation for the piecewise function to the left, then walk through how we would graph it, and then determine the domain and range.
Let's begin with the parabola since it is furthest left. The parabola is the parent parabola y=x². But, the parabola ceases to exist at x=2. So the parabola would be denoted as y=x², x<2.
The line is horizontal and at y=-1 and beings to exist at x=2, so it is denoted as y=-1, x<2.
Our final equation is
f(x) {x², x<2
-1, x>2
Notice that we use a "greater than or equal to" sign for the horizontal line's domain. The "equal to" indicates that the line y=-1 will have a closed dot at 2. But, y = x² has just a "less than" sign which indicates that the circle at 2 will be open. This is very important as it makes the function a function.
Graphing is simple, just work backwards. If we couldn't already see the graph, we would begin by examining the equation. We would draw the parabola first and then the line. Then, we would erase any of the parabola showing after x=2. We would put an open dot at x=2 for the parabola and an arrow on the other end. The line we would erase after x=2. We would put a closed dot at x=2 and an arrow on the other end. We would then label the pieces of our piecewise function. The graph above is missing labels and arrows.
Let's determine the domain of the piecewise function. We can see that the function goes all the way to negative infinity so: (-infinity. (Remember that infinity is not obtainable so we use a parenthesis). The function also goes on to positive infinity, so our final domain would be: (-infinity, infinity).
Try finding the domain of the first graph on the right. Click the button below to reveal the answer.
Let's begin with the parabola since it is furthest left. The parabola is the parent parabola y=x². But, the parabola ceases to exist at x=2. So the parabola would be denoted as y=x², x<2.
The line is horizontal and at y=-1 and beings to exist at x=2, so it is denoted as y=-1, x<2.
Our final equation is
f(x) {x², x<2
-1, x>2
Notice that we use a "greater than or equal to" sign for the horizontal line's domain. The "equal to" indicates that the line y=-1 will have a closed dot at 2. But, y = x² has just a "less than" sign which indicates that the circle at 2 will be open. This is very important as it makes the function a function.
Graphing is simple, just work backwards. If we couldn't already see the graph, we would begin by examining the equation. We would draw the parabola first and then the line. Then, we would erase any of the parabola showing after x=2. We would put an open dot at x=2 for the parabola and an arrow on the other end. The line we would erase after x=2. We would put a closed dot at x=2 and an arrow on the other end. We would then label the pieces of our piecewise function. The graph above is missing labels and arrows.
Let's determine the domain of the piecewise function. We can see that the function goes all the way to negative infinity so: (-infinity. (Remember that infinity is not obtainable so we use a parenthesis). The function also goes on to positive infinity, so our final domain would be: (-infinity, infinity).
Try finding the domain of the first graph on the right. Click the button below to reveal the answer.
Note that piecewise functions CAN have a break in the graph and can still be a function. Examine the graph to the left. The function does not exist at x=0 because both rays have an open dot. However, this is still a function because technically, each x input still has one y output.
This function is just discontinuous, meaning that there is a break. The two piecewise functions above are continuous because there are no breaks in the functions. However, both discontinuous and continuous piecewise functions are functions.
The domain for the piecewise function to the left will use a "u" union symbol to represent that the domain is for a function, but there is a break: (-infinity, 0) u (0, infinity).