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### Course: Get ready for AP® Calculus > Unit 1

Lesson 4: Piecewise functions# Worked example: evaluating piecewise 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 evaluate piecewise functions (find the value of the function) by using their formulas or their graphs.

## Want to join the conversation?

- What is the E symbol mean? (1:25)(25 votes)
- The symbol ∈ indicates set membership and means “is an element of”.

I hope this has been helpful.(25 votes)

- You matched g(4.0001) with -3 open circle but you didn't match g(9) with 3 you said because its a open circle and there is no closed circle so it is undefined why ?(9 votes)
- For this third piece of the piece wise function, we have that when 4 < x < 9, then f(x) = 3. Notice that x cannot be equal to 4 or 9; it has to be greater than 4 and less than 9.

If they had asked for g(4), that would be undefined to since the open circle on 4 means that the value 4 is NOT included. BUT any value ever so slightly greater than 4 IS included, so g(4.0001), which is 0.0001 greater than 4, so it IS included.

Now when x=9, f(x) is undefined. BUT if we got close to x=9, say x=8.999, then that would be defined and g(8.999)=-3.(19 votes)

- Brackets mean included and parenthesis mean up to the number but not including it, correct? Or have I learned them in the wrong order?(14 votes)
- For designating intervals, you've got it exactly right.(6 votes)

- someone please explain what empty and filled circles mean(5 votes)
- Empty and filled circles tell you whether a value is included or not.

Empty circle = Used for < and >

Filled circle = Used for ≤ and ≥

For example, at0:28, the instructor says t is less than or equal to -10 in the first function. Therefore, you plot a full circle at the point where t = -10 and graph the function for the values less than -10 from there.

On the other hand, the second function is for values -10 < t < -2. This means you plot an empty circle at the point where t = -10 and an empty circle at the point where t = -2. You then graph the values in between.

Finally, for the third function where t ≥ -2, you plot the point t = -2 with a full circle and graph the values greater than this.

When you combine all three pieces, you can see the piecewise function at02:27.

Hope this helps!(12 votes)

- At1:45, which case would we use if we wanted to find out h(0) ?(4 votes)
- You would use the 1st option. The domains for each piece are defined using interval notation. Since the 1st piece is defined for x = (-infinity, 0], the square bracket tells us that this is <=0. On the next piece, you will see x = (0, 8]. The parentheses on the 0 tell us that zero is not in this domain, while the square bracket on the 8 tells us the 8 is in the domain for this piece.

Hope this helps.(7 votes)

- What does the element of sign mean (E) and whats the difference between it and the less than, greater than, less than or equal to, and greater than or equal to signs? At1:23(4 votes)
- The domain for each piece is being shown in interval notation. Each interval represents a set of continguous real numbers. And the element of symbol is saying the value must be in that set. You can convert between interval notation and inequalities. For example: x ε (-infinty, 0] is the same as x<=0(6 votes)

- What are some real-world scenarios that can be modeled by a piece-wise function?(2 votes)
- Postage is often piecewise, the cost depends on the weight in ranges. If you buy products from a company, they often charge shipping costs according to how much you spend (and these are in a range of numbers). Plumbers and other salaries are often piecewise because they will charge for the full hour for any part of the hour the work, so it steps up by the hour.(9 votes)

- Why is g(9) undefined? Why isn't it -3?(2 votes)
- There is an open dot where x=9. It means that the graph goes up to 9 but does not include the 9. Since 9 is excluded for this function, it is undefined.(8 votes)

- if g(9) is undefined, shouldn't g(4.0001) also be? it's an open dot and also the same line.(3 votes)
- The open dot is at (4, -3). Since 4.001 is to the right of 4, it is on the line and is a defined value for the function.

Hope this helps.(4 votes)

- in t^2-5t why (-10)^2 and not -10^2?(4 votes)
- In the first correct method, you are squaring the number -10

In the second method, you only square 10 and changing the sign

The importance is that a negative times a negative is a positive, so squaring any negative number will always give you a positive

And later on, you will find that you cannot take the square root of a negative number unless you go into a part of Math called imaginary numbers(2 votes)

## Video transcript

- [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. And 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. So 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. - 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. - 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. So, 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. If 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. And 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. It 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. g(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. I 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. So 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. So 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.