Piecewise Functions: Videos & Practice Problems

Understanding piecewise functions is essential in mathematics, as they consist of multiple equations defined for specific intervals of x values. To graph a piecewise function, identify the boundaries where the equations change, ensuring no overlap occurs. For example, if x is less than -1, use one equation, while for x greater than or equal to -1, use another. Evaluating these functions involves substituting x values into the appropriate equations, which may reveal discontinuities, such as jump discontinuities, indicating that the function is not continuous across its domain.

Piecewise Functions

One of the things we should be very familiar with is the concept of a function. We've talked about these functions and how they are represented as equations, and we've also talked about how we can take these equations and graph them individually. Now, what we're going to be learning in this video is how we can take multiple equations like this and combine them into a single function. This is known as a piecewise function, and a piecewise function is made up of multiple equations. Because we cannot have a situation where these equations overlap each other, what we need to do is define these equations for different x values when dealing with piecewise functions.

So let's just get right into an example of a piecewise function situation we would need to solve in this course. Let's say we're dealing with this function right here. This would be an example of a piecewise function because notice how we have multiple equations written for a single function and how we define the equations for different x values. Now, I need to take a look at these x values and figure out how I can graph this equation.

Well, the way that I can graph this function, f of x, is I notice here that our x values are really interested in negative one. Because notice we're looking where $x$ is less than negative one or where $x$ is greater than or equal to negative one. I'll pause right here and mention that it's possible you also have more than 2 equations. You might have 3 or 4 equations where more x values are defined. But since here we're only looking at negative one, I'm going to draw a wall right here through this x value of negative one.

This wall tells me where my functions are going to be defined. So everything on the left side of the wall will be this equation, and everything on the right side of the wall will be that equation. This equation shows us right here that we have $x\ge -1$, whereas the other equation shows us where $x<-1$. And notice how these are the same equations that we had over here. So if I were to draw a wall on my graph, what I'm doing is taking this left piece of the equation, drawing on the left side of the wall, then taking this right piece of the equation and drawing it on the right side of the wall.

We first will graph $-x$ where $x<-1$. This will be the first piece of our piecewise function. The next piece is where $x\ge -1$. We'll notice that it's a parabola. However, I can't just draw a full parabola because we are only defining the parabola on the right side of this wall. When we have this negative one value, and we're down here at negative 3, we'll have a parabola that looks something like this, and this would be the right side of this piecewise function.

I'll now go ahead and remove this wall because this is what the piecewise function is going to look like. But what I also need to do, because I'm not quite finished, we have these empty points finishing here, and we need to write something for these. For this piece saying $x<-1$, it can't actually be equal to negative one, so we have to put an open circle right there. For this other piece, we see that $x\ge -1$, so I can put a solid circle right there.

These two parts are not connected. There's a jump that happens right here. Whenever you have the situation where the y values between the pieces don't match, we call this a jump discontinuity, because we have this jump right here. It is not going to be continuous. I have to pick up my pen and move to another part of the graph. This jump indicates that the curve is not continuous, but that's okay because this happens a lot when dealing with these piecewise functions.

We also need to evaluate these functions by plugging values into the correct equations. My first function that I'm trying to evaluate at an x value of negative 3, which is clearly less than negative one, falls into this portion of the equation. So I'm going to use negative x, giving us positive 3. This right here would be $f(-3)$. Now, for $f(-1)$, notice negative one fits here because $x\ge -1$. We'll input -1 into this equation giving us negative 3 again, so this would be $f(-1)$. For $f(2)$, $x$ is greater than -1, so we use the equation $x^2-4$, giving 0. This is $f(2)$.

Now that we found these evaluations by plugging them into our piecewise function, you can take a look at graph and notice that all of these points line up to where we said they would based on the graph we have. This is how you can graph piecewise functions and also evaluate them using these equations.

I hope you found this video helpful. Let's go ahead and try getting a little more practice with this concept. See you in the next one.

Using the piecewise function below, evaluate $$$f\left(-1\right)$

• A. $-4$
• B. $4$
• C. $\pm \sqrt{3}$
• D. $2$

Answer: (B)

Using the piecewise function below, evaluate $f\left(5\right)$

• A. $130$
• B. $125$
• C. $120$
• D. $30$

Answer: (A)

Piecewise Functions Example 1

In this problem, we're asked to graph the following piecewise function. To graph this function, we first need to identify the boundaries on our graph. For the first part of the function, we note that x is less than negative 2, so we draw a line at x equals negative 2 to represent this boundary.

Next, the function specifies that x is between negative 2 and 1. Thus, we draw another line at x equals 1. The function for values of x greater than or equal to 1 is represented to the right of this line. As such, the function is divided into three distinct parts between these boundaries.

Starting with the first segment of the equation, it states that the function is equal to negative 4, a constant value. Graphically, this is represented as a horizontal line at y equals negative 4, extending indefinitely to the left but stopping at x equals negative 2.

Moving on to the second part, the function is described by x plus 1. This linear function intersects the y-axis at 1 and continues to increase. This line will only exist between x equals negative 2 and x equals 1, constrained by the defined boundaries.

For the final part, the function is x squared, which forms a parabola. This part of the function starts at x equals 1, where the value of the function is 1 (since 1 squared equals 1), and extends indefinitely to the right, increasing in value.

When considering where to place dots or open circles to indicate boundaries of definition, observe the inequalities defining each segment. For the first segment (x less than negative 2), we place an open circle at x equals negative 2. For the line segment defined from negative 2 to 1, we place a solid dot at x equals negative 2 and an open circle at x equals 1 because x is strictly less than 1. For the final segment starting at x equals 1, a solid dot at this point indicates that x equals 1 is included in the domain of this part of the function.

This visualization helps to understand how the piecewise function is structured and ensures clarity in its graphical representation. Such practice is essential for mastering the graphing of complex functions.

Here’s what students ask on this topic:

What is a piecewise function and how is it defined?

A piecewise function is a function composed of multiple sub-functions, each defined over a specific interval of the domain. It is represented as:

$f\left(x\right)=\{\begin{array}{cc}f\_1\left(x\right)& if x < a\\ f\_2\left(x\right)& if a \le x < b\\ f\_3\left(x\right)& if x \ge b\end{array}$

Each sub-function applies to a specific interval of x values, ensuring no overlap between intervals.

How do you graph a piecewise function?

To graph a piecewise function, follow these steps:

1. Identify the intervals for each sub-function.
2. Draw vertical lines (walls) at the boundaries of these intervals.
3. Graph each sub-function within its respective interval, ensuring to use open or closed circles at the boundaries based on the inequality signs.
4. Check for discontinuities, such as jump discontinuities, where the function values do not match at the boundaries.

This method ensures that each piece of the function is accurately represented within its defined interval.

What is a jump discontinuity in a piecewise function?

A jump discontinuity occurs in a piecewise function when there is a sudden change in the function's value at a boundary point. This means the left-hand limit and the right-hand limit at the boundary do not match. For example, if:

$f\left(x\right)=\{\begin{array}{cc}-x& if x < -1\\ \text{msup}& if x \ge -1\end{array}$

There is a jump discontinuity at x = -1 because the function values on either side of -1 do not match.

How do you evaluate a piecewise function at a specific x value?

To evaluate a piecewise function at a specific x value, follow these steps:

1. Determine which interval the x value falls into.
2. Use the corresponding sub-function for that interval.
3. Substitute the x value into the chosen sub-function and solve.

For example, if:

$f\left(x\right)=\{\begin{array}{cc}-x& if x < -1\\ \text{msup}& if x \ge -1\end{array}$

To find f(-3), use the first sub-function: f(-3) = -(-3) = 3.

Can a piecewise function be continuous?

Yes, a piecewise function can be continuous if the function values match at the boundaries of the intervals. This means the left-hand limit and the right-hand limit at each boundary point must be equal. For example, if:

$f\left(x\right)=\{\begin{array}{cc}-x& if x < -1\\ \text{msup}& if x \ge -1\end{array}$

To ensure continuity at x = -1, the function values from both sub-functions must be equal at x = -1.