Worked example: evaluating piecewise functions

0:00- [Instructor] Consider the following piecewise function and we say f(t) is equal to and they tell us what it's equal to based on what t is, so if t is less than or equal to -10, we use this case. If t is between -10 and -2, we use this case. And if t is greater than or equal to -2, we use this case.

0:18And then they ask us what is the value of f(-10)? So t is going to be equal to -10, so which case do we use? So let's see. If t is less than or equal to -10, we use this top case, right over here and t is equal to -10, that's the one that we're trying to evaluate.

0:34So we wanna use this case right over here. So f(-10) is going to be equal to -10, everywhere we see a t here, we substitute it with a -10.

0:48- 10 squared minus 5 times, actually I don't have a denominator there, I don't know why I wrote it so high. So it's gonna be -10 squared minus 5 times -10. So let's see.

1:01- 10 squared, that's positive 100 and then negative, or subtracting 5 times -10, this is going to be subtracting -50 or you're going to add 50, so this is going to be equal to 150. f(10) is 150, 'cause we used this case up here, 'cause t is -10. Let's do another one of these examples.

1:25So, here we have consider the following piecewise function, alright. What is the value of h(-3)? See when h is -3, which case do we use? We use this case if our x is between negative infinity and zero. And -3 is in between negative infinity and zero, so we're gonna use this case right over here.

1:48If it was positive three, we would use this case. If it was positive 30, we would use this case. So we're going to use the first case again and so for h(-3), we're gonna take -3 to the third power. So let's see. h(-3) is going to be -3 to the third power which is -27.

2:11And we're done. That's h(-3). Because we're using this case, you could almost ignore these second two cases right over here. Let's do one more example. This one's a little bit different. Below is a graph of the step function g(x) so we can see g(x) right over here.

2:28It starts when x equals -9, it's at 3, and then it jumps up, and then it jumps down. Match each expression with its value. So g(-3.0001), so -3.0001, so that's right over here and g of that, we see is equal to 3. So this is going to be equal to 3 right over here.

2:54g(3.99999) 3.99999, almost 4, so let's draw a dotted line right here, it's gonna be almost 4, well g(3.99999) is going to be 7. We see that right over there. So that is equal to 7. g(4.00001). So g(4) is still 7, but as soon as we go above 4, we drop down over here, so g(4.00001) is going to be -3.

3:29I wanna, actually, let's focus on that a little bit more. How did I know that? Well I know that g(4) is 7 and not -3 because we have this dot is circled in up here and it's hollow down here. But as soon as we get any amount larger than 4, then the function drops down to this.

3:46So 4.0000, as many, just slightly above 4, the value of our function is going to be -3. Now let's do g(9). So g(9), that's when x is 9 and we go down here. You might be tempted to say it's -3, but you see, at this point right over here, we have an open circle.

4:05So that means that while it's not, you can't say that the function is -3 right over there and there's no other place where we have a filled-in circle for x equals 9 so the function g actually isn't defined at x equals 9. So I'm gonna put undefined right over there.