If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 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? ()
• The symbol ∈ indicates set membership and means “is an element of”.
I hope this has been helpful.
• Brackets mean included and parenthesis mean up to the number but not including it, correct? Or have I learned them in the wrong order?
• For designating intervals, you've got it exactly right.
• 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 ?
• 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.
• someone please explain what empty and filled circles mean
• 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, at , 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 at .

Hope this helps!
• At , which case would we use if we wanted to find out h(0) ?
• 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.
• What are some real-world scenarios that can be modeled by a piece-wise function?
• 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.
• in t^2-5t why (-10)^2 and not -10^2?
• 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
• can you explains this word mean please "∈"
• The "∈" is used in set notation.
If you see something like: "x ∈ Integers", then it is telling you that "X is an element or member of the set of integers".
Hope this helps.
• What's the explanation/convention behind a parenthesis for infinity ∞ in a set? In other words, why is it (-∞, 5] or [-1, ∞)? Why not [-∞, 5] or (-1, ∞]?
• The square bracket is only used then the interval includes a specific number, like the 5 or the -1. It says that specific value is included in the interval. Since infinity is not a specific number and represents an infinite set of values, we always use a parentheses.

Hope this helps.