- Introduction to piecewise functions
- Worked example: evaluating piecewise functions
- Evaluate piecewise functions
- Evaluate step functions
- Worked example: graphing piecewise functions
- Piecewise functions graphs
- Worked example: domain & range of step function
- Worked example: domain & range of piecewise linear functions
A piecewise function is a function that is defined in separate "pieces" or intervals. For each region or interval, the function may have a different equation or rule that describes it. We can graph a piecewise function by graphing each individual piece.
- [Voiceover] So, I have this somewhat hairy function definition here, and I want to see if we can graph it. And this is a piecewise function. It's defined as a different, essentially different lines. You see this right over here, even with all the decimals and the negative signs, this is essentially a line. It's defined by this line over this interval for x, this line over this interval of x, and this line over this interval of x. Well, let's see if we can graph it. I encourage you, especially if you have some graph paper, to see if you can graph this on your own first before I work through it. So, let's think about this first interval. When negative 10 is less than or equal to x, which is less than negative two, then our function is defined by negative 0.125x plus 4.75. So this is going to be a line, a downward sloping line, and the easiest way I can think about graphing it is let's just plot the endpoints here, and then draw the line. So, when x is equal to 10, sorry, when x is equal to negative 10, so we would have negative zero, actually let me write it this way. Let me do it over here where I do the, so we're going to have negative 0.125 times negative 10 plus 4.75. That is going to be equal to, let's see, the negative times a negative is a positive, and then 10 times this is going to be 1.25 plus 4.75. That is going to be equal to six. So, we're going to have the point negative 10 comma six. And that point, and it includes, so x is defined there, it's less than or equal to, and then we go all the way to negative two. So, when x is equal to negative two, we have negative 0.125 times negative two plus 4.75 is equal to, see negative times negative is positive, two times this is going to be point, is going to be positive 0.25 plus 4.75. It's going to be equal to positive five. Now, we might be tempted, we might be tempted, to just circle in this dot over here, but remember, this interval does not include negative two. It's up to and including, it's up to negative two, not including. So, I'm gonna put a little open circle there, and then I'm gonna draw the line. And then I'm gonna draw the line. I am going to draw my best attempt, my best attempt, at the line. Now, let's do the next interval. The next interval, this one's a lot more straightforward. We started x equals negative two, when x equals negative two negative two plus seven is, negative two plus seven is five. So, negative two, so negative two comma five, so it actually includes that point right over there. So we're actually able to fill it in, and then when x is negative one, negative one plus seven is going to be positive six. Positive six, but we're not including x equals negative one up to and including, so it's going to be right over here. When x is negative one, we are approaching, or as x approaches negative one, we're approaching negative one plus seven is six. So, that's that interval right over there. And now let's look at this last interval. This last interval, when x is negative one, you're going to have, well, this is just going to be positive 12 over 11 'cause we're multiplying it by negative one, plus 54 over 11 which is equal to 66 over 11 which is equal to positive six. So, we're able to fill in that right over there, and then when x is equal to 10, you have negative 120 over 11. I just multiplied this times 10, 12 times 10 is 120, and we have the negative, plus 54 over 11. So this is the same thing. This is going to be, what is this? This is negative 66 over 11, is that right? Let's see, if you, yeah, that is negative 66 over 11, which is equal to negative six. So when x is equal to 10, our function is equal to negative six. And so this one actually doesn't have any jumps in it. It could've, but we see, so there we have it. We have graphed this function that has been defined in a piecewise way.