[MUSIC] Let's look at piecewise-defined functions.

A piecewise-defined function is a function that is defined in pieces or

according to different rules depending upon the input.

[SOUND] So for example, this would be considered a piecewise-defined function.

So what does this mean? It means that f is defined by different rules according

to what x is. That is, if x does not equal -1, then

f(x) = -3. Otherwise, if x does equal -1, then f(x)

is equal to -4. So, there are different rules that define

f depending upon the input x. So let's compute f(-5), f(-1), and f(2)

and then we'll graph f. Let's start with the f(-5).

[SOUND] Since -5 does not equal -1, we'll be using this first piece here.

That is f(-5) = -3. Alright,

and what about f(-1)? [SOUND] We'll be using the second piece down here,

because the input is -1, which means f = -4.

And finally, what about f(2)? [SOUND] Well, since 2 is not equal to -1, we'll

be using this first piece. That is f(2) = -3.

So, it remains for us now to graph f. So, let's say that this is the y-axis and

this is the x-axis, and let's say this is -1, and this is -3, and this is -4.

As long as x does not equal -1, f(x) or y = -3 and y = -3 is the equation of this

horizontal line here. However, when x = -1, we have an open

circle, because when x = -1, the y value is -4.

So this would be the graph of f. Let's look at another example.

[SOUND] Let g be defined by this piecewise function.

We're going to find g(-5), g(-2), and g(2) and then we'll graph g.

So let's start with computing g(-5). [SOUND] Now, which of these three

intervals does -5 lie in? It lies in this first interval, doesn't it? Because -5 <

-3, therefore, we're going to use the first

rule or the first piece to compute this. So g(-5) = -4.

Alright, what about g(-2)? Well, which piece or rule are we going to use to

compute this? [SOUND] Well, -2 lies in the second interval, doesn't it? So we'll

use the second rule. That is, g(-2) = -2 -1 which is equal to

-3. And finally, what about g(2)?

[SOUND] Which of the three intervals does 2 lie in? And we have to be careful here,

because we see a 2 here and we see a 2 here.

However, the condition of equality is down in this third interval here.

Therefore, we use this third piece. That is, g(2) = 3.

Okay. So it still remains to graph g. So let's say this is the y-axis and this

is the x-axis, and let's say this is -3, and this is +2, and this is +3, and -4.

When x is strictly less than -3, then y or g(x) = -4.

Therefore, any x less than -3, y = -4. And then, for any x between -3 and 2, the

function is defined to be x - 1, which is the equation of the line with y

intercept equal to -1 and slope 1. So it will look like this.

And right when we get to two, we have an open circle, because looking back over

here, this is a strict inequality. But also looking back over here, we have

the condition of equality at -3, which means, over here we have to close

this circle up. And also, this will be negative one

because the y intercept is negative one and this would be one because the slope

is one. And, then, for x > 2,

g(x) = 3. So, we have a closed circle at 2,

and then, y = 3 is the equation of a straight line.

And this is how we work with piecewise-defined functions.

Thank you and we'll see you next time.

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