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## Algebra (all content)

### Course: Algebra (all content)>Unit 7

Lesson 25: Graphing nonlinear piecewise functions (Algebra 2 level)

# Graphs of nonlinear piecewise functions

Sal is given the graph of a piecewise function and several possible formulas. He determines which is the correct formula. Created by Sal Khan.

## Want to join the conversation?

• How can a certain graph 'look like' a certain function? Is this something you have to know by heart or start recognizing over time?
EDIT: thanks for all the help. I'd just like to add that one can use Desmos Graphing Calculator (online, free) to easily draw these graphs. • if the graph was shifted to the left 2, then why is the graph sqrt(x +2)? similarly if the graph was shifted to the right 4, then why is the graph (x-4)^3? • where can i find practice problems for this? • So what is the official definition of a piece-wise function and how can they be applied to the real world? • Does anyone have a method for determining this answer quicker? I know that there are questions like these on tests, and Sal seems to be testing all of the options, but what if the correct answer was the last one? The method seems to be a bit time-consuming, and I just wanted to know if anyone had any tips. • Yes. If you scan down the answers, you will see that all of them describe two sub-functions, and in every case the sub-functions are of a similar form, i.e. based on √x and x³.
Therefore, the answer must include a √x and a x³ type function over parts of the domain of x.
You can also see that, in the case of the √x type function, each answer is dissimilar from the others. In each case, either the form of the function, or the interval over which it is defined is different from the others. This means that you need only concentrate on the √x type sub-functions to identify the correct answer.
You may probably recognise the shape of the graph of a √x function, but even if you had never seen one, you know for example that the function will not increase as steeply as x³ for large values of x (those greater than one), and that it will be undefined for negative values of x. This will tell you that you need to focus on the graph that you see defined on the interval -2 < x <= 2 (see as in see the picture of the graph and read off the interval from the x-axis)
Only f(x) and h(x) are defined over this interval with a √x type sub-function, so if you can distinguish between √(x-2) and √(x+2), you wll know the answer.
You can know immediately that when x=2, f(x) will be zero, but h(x) will be √4. Looking at the graph, the graphed function is not zero when x = 2, so h(x) must be the answer.
You could also look at the points where the graphed function has a zero value, at x= -2 and x=4 (although x = -2 is technically just beyond the boundary of the domain of x), and see which of the answers would provide those zero values. Only p(x) and h(x) do so, but we have established that p(x) isn't defined over the correct interval for the √x type sub-function.
• what does it mean when the point is an open circle? • What exactly is a piecewise function? When would it be used in real life?
(1 vote) • Why is it (x-4)^3 instead of (x+4)^3? • What does it mean if a point is an open circle? • If a point is an open circle, then it is not included in the function. Here, there are two open circles. These circles signify that the function goes through or to there somehow so everything before or after it is part of the function, but the open circle itself is not part of the function. (-2, 0) and (2, -8) are not part of this function since there are open circles there, but the blue lines after those points are part of the function.
• I'm having a lot of problems with this. How am I supposed to know what a function looks like. I've been doing KH since Algebra and I'm confused. I watched the videos in this section. Did a miss a video where it explains what functions are supposed to be look like graphed. Please assist.

Thanks 