Intro to Functions & Their Graphs - Video Tutorials & Practice Problems

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1

concept

Relations and Functions

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Hey, everyone. So up to this point, we've spent a lot of time talking about graphs. And in this video, we're going to see if we can apply this concept of graphs to this new topic of relations and functions. Now, this topic is often considered confusing when students initially encounter it. But throughout this video, we're going to be going over a lot of different scenarios and examples to see if we can really clear up some of this confusion around this subject. So let's get right into this relations are a connection between X and Y values and graphically they are represented as ordered pairs. Now, functions are a special kind of relation where each input has at most one output. And it's important to note that all functions are relations but not all relations are functions. So to understand this a little bit better, let's take a look at this example that we have down here, we have these two graphs of relations and we want to see if we can determine whether these are also examples of functions. We'll start with this graph on the left. And what I'm going to do is write out all of the inputs which correspond to all of the X values. So I'll do in ascending order. So first, I see that we have an X value of negative two. I also see that we have an X value of positive one. And even though it shows up twice here, it's perfectly OK to only write it once in this bubble down here. Now, lastly, I see that we have an X value of three. So these are all of the inputs that we have. Now the outputs are going to correspond to the Y values and I'll list all of these out as well. So I see that we have positive two. I see that we have four, we also have one and then we have negative two. So these are all of the inputs and outputs. Now looking at these points, I see that negative two is related to positive two. I see that positive one is related to four and I also see that one is related to one. Now, lastly, I see that we have this point which says three is related to negative two. Now, based off this relation that we see, can we conclude whether or not it's a function? Well, recall that we set up here for a relation to be a function each input can have at most one output. And if I look at each of our inputs, I can see that there is an input that has more than one output. And as soon as this happens, you can automatically conclude that this is not an example of a function. But let's take a look at this other example for this graph on the right. So what I'm gonna do with this graph is I'm going to list out all of the inputs in ascending order like we did before. So this will be all of the X value. So I see that we have negative four, I see that we have negative two. I see that we have one and then we have three. Now what I'm also going to do is list out all of the outputs which correspond to all the Y values. So I see that we have positive two. I see that we have negative one and I also see that we have positive two up here. But since we already wrote positive two, once we don't have to write it again. So we have positive two and then we have positive four. So these are all of the outputs. Now looking at how these are related, I can see that negative four is related to make or to excuse me to positive two. I see that negative two is related to negative one and I see that positive one is related to positive two. And then I see that we have that positive three is related to positive four. Now given this information, can we conclude whether or not this is a function? Well, we need to see if any of the inputs have more than one output. And if I look at this, each of these inputs only go to one output. And because of this, we would say this is an example of a function. So this is how you can tell whether or not a relation is a function. But you may have noticed this process was a bit tedious having to write out all the inputs and outputs like this. Well, you may be happy to know there is a shortcut to solving these problems. And the shortcut is called the vertical line test. This states. If you can draw any vertical line that passes through more than one point on your graph, then the graph is not going to be a function. So let's try this vertical line test on the two graphs we had up here, I'll take vertical lines and I'll draw them through every point that I see on the graph. And let me draw this vertical line a bit better. So if I draw these vertical lines, I noticed that there is a place where the vertical line passes through more than one point. If this ever happens, then it's not a function. But let's try the vertical line test on this other graph. If I draw vertical lines through each of the points that I see, I notice that no matter where I draw a vertical line, I'm only ever gonna pass through one point at most. And because of this, we would say that this is an example of a function. Now let's take a look at a couple more examples because you could also use the vertical line tests on graphs like this. If I tried drawing some vertical lines for this graph on the left, I noticed that we will have some vertical lines that pass through more than one point. And because of this, we can conclude that this is not an example of a function. But if I try the same vertical line test on the other graph that we have over here, notice that no matter where I draw a vertical line, we're only ever going to pass through one point at most. And because of this, we can conclude that this graph is an example of a function. So that's the basic idea of relations and functions. Hopefully, this helped you out and let me know if you have any questions.

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Problem

Problem

State the inputs and outputs of the following relation. Is it a function? {$\left(-3,5\right),\left(0,2\right),\left(3,5\right)$}

A

B

C

D

3

Problem

Problem

State the inputs and outputs of the following relation. Is it a function? {$\left(2,5\right),\left(0,2\right),\left(2,9\right)$}

A

B

C

D

4

example

Example 1

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So let's see if we can solve this problem in this problem, we have three different graphs and we're asked to determine below which of the graphs are functions and we can select all that apply. Now whenever you want to figure out whether or not a graph is a function, you can use the vertical line test. If you can draw a vertical line somewhere on the graph that crosses through more than one point, it is not a function. However, if the vertical line you draw only can cross through one point at most anywhere you draw it, it is a function. So let's see if we can figure this out. Now, for this first graph, we'll try the Fortical line test. If I draw a vertical line here, we only cross through one point. If I try a vertical line over here, we only cross through one point. If I try a vertical line through the center or close to the center, we only cross through one point. And it turns out even though this graph expands in both directions, anywhere you would draw a vertical line would only ever cross through one point. So we can see that this first graph graph A is an example of a function. But let's try graph B over here. And I'll go ahead and draw a vertical line. Notice the vertical line we drew crosses through more than one point since we see that we can draw a vertical line somewhere that crosses through more than one point. That means this is not an example of a function. But now let's try the vertical line test on this graph over here. Well, if I draw a vertical line there, we only cross through one point. If we draw a vertical line here, only cross through one point. And anywhere I would draw this vertical line would only ever cross through one point. So we would say that graph C is an example of a function. So to list all of the graphs that are functions, we would say graph A and graph C are functions. That's the answer to this problem.

5

concept

Verifying if Equations are Functions

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Welcome back everyone. In the last video, we talked about relations and functions and we specifically focused on graphs. Now recall in the previous video that we had this special tool called the vertical line test where we could see if a vertical line ever passed through more than one point anywhere we would draw it on the graph. If this ever happened, we would say that we failed the vertical line test. In which case, we would not be dealing with a function. However, if this did not happen, then we were dealing with a function. We talked about all of this in the last video. Now in this video, we're going to be taking a look at how we can verify equations as functions and there is a test for this. But the steps are a little bit more complicated. So pay close attention because I am going to walk you through them. The first thing you want to do when dealing with an equation is always solve for Y this should be your first step when you have an equation. So if you have an equation and you've solved for Y what you then want to do is see if any X's will result in multiple Ys. If this does happen, then you are not dealing with a function. However, if this does not happen, then you are dealing with a function. Now notice for both of these tests, whenever we get an answer of yes, that means it's not a function. But whenever we get an answer of no, it is a function. So that's just something you can use to remember how these tests work. Now, let's actually put this test to use and solve an example. So here we have these two equations and we want to see whether or not either of these equations are functions. We'll start with equation A which is this one on the left, we have Y plus four is equal to three X. And our first step should always be to solve for Y. So I'll take this four and subtract it on both sides of the equation to cancel the fours on the left. That will give me that Y is equal to three, X minus four. And now that we've done this, we have solved for Y. So the next thing I'm gonna do is I'm going to try a bunch of X values to see if we have a situation where any of these X's result in multiple Ys. So let's try an X value of, let's just start with zero. So we'll have that Y is equal to three times zero minus 43 times zero is zero and zero minus four is negative four. Now let's try an X value of one. We have Y is equal to three times one minus 43 times one is three and three minus four is negative one. Now let's try a negative number like negative one. So we'll have Y is equal to three times negative one minus 43 times negative one is negative three and negative three minus four is negative seven. Now let's try an X value of two. We'll just try one more value here to see what happens. So we have three times two minus 43 times two is six and six minus four is two. Now notice what happens here for every X value that we replace this X with, we always get only one Y value as an output. And because of this, we could say that this is an example of a function because we only get one Y value for each X value. And you may notice that this is actually a familiar form that we have this equation. We have the form Y equals MX plus B. This is the slope intercept form of a line that we've talked about in previous videos. So whenever you are dealing with a line that has a defined slope, and in this case, the slope would be positive three, then you're always going to be dealing with a function in this situation. So that's just something you can remember when dealing with these types of problems. But now let's take a look at this equation on the right side. So for equation B, we have X squared plus Y squared equals 25. And just like before I'm gonna solve for Y, so I'll subtract X squared on both sides of the equation. And that'll get the X squared to cancel on the left giving me that Y squared is equal to 25 minus X squared. Now, my next step here is going to be to take the square root on both sides of the equation giving me that Y is equal to plus or minus the square root of 25 minus X squared. So we've now solved for Y. So this is the equation that we have. Now, our next step is going to be to test out a bunch of different X values. So just like before I'll start with an X value of zero, so we'll get that Y is equal to plus or minus the square root of 25 minus zero squared. Zero squared is just zero. So all we're gonna end up with is plus or minus the square root of 25. And we're called the square root of 25 is just five. So because of this plus or minus sign, we're going to end up with positive five and negative five. But this is interesting notice how one X value gave us multiple Y values and we said whenever this happens that we are not dealing with a function. So in this example, we can see that this is not a function because we already have an X value that gives us multiple Y values. And something else you can remember with this equation is notice that we started with Y squared. Whenever you have an equation where Y has an even power that it's raised to, then it's always going to be not a function. So that's just something to keep in mind. Now, neither one of these situations are scenarios that you're going to have to memorize. But it's just something that can help you if you want a quick shortcut to figuring out whether an equation is a function. Now, there's one more thing I want to mention before finishing this video and that is if you have an equation that is identified as a function, you can rewrite it using function notation. And when you use function notation, what you want to do is replace the Y with F of X. So for example, in this scenario that we had before where we had Y equals three, X minus four. So this equation up here, you could take this Y at the start and you could replace it with F of X where X represents the inputs and F of X represents the outputs. You could not however use this function notation in the other example, because this was not identified as a function. We actually figured out that it's not a function. So this is how you can verify equations as functions. Let me know if you have any questions and let's move on.

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Problem

Problem

Is the equation $y=-2x+10$ a function? If so, rewrite it in function notation and evaluate at $f\left(3\right)$.

A

$f\left(3\right)=4$ , Is A Function

B

$f\left(3\right)=3$, Is A Function

C

$f\left(3\right)=1$, Is A Function

D

Is NOT A Function

7

Problem

Problem

Is the equation $y^2+2x=10$ a function? If so, rewrite it in function notation and evaluate at $f\left(-1\right)$.

A

$f\left(-1\right)=\sqrt{12}$, Is A Function

B

$f\left(-1\right)=12$, Is A Function

C

$f\left(-1\right)=\frac92$, Is A Function

D

Is NOT A Function

8

concept

Finding the Domain and Range of a Graph

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5m

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Hi, everyone. And welcome back up to this point, we spent a lot of time talking about functions. And in this video, we're going to be taking a look at how we can find the domain and range of a function based on its graph. Now, when encountering finding the domain and range of functions, it can often seem a bit tricky. But in this video, we're going to be going over some analogies and examples that will hopefully make this topic a lot more clear. So let's get into this when finding the domain of a graph, you're looking for the allowed X values that you can have and when finding the range you're looking for the allowed Y values. Now, there's a little trick that you can use and that is known as the squish strategy. What you can do if you have the domain of a graph is you can take your graph and squish it down to the X axis. So if you squish things down to the X axis, that will tell you the domain. So let's say we have this graph down here and notice how we took this curve and we literally squished it down to the x axis. So only some sort of line or set of lines remains, this tells you the domain. Now when finding the range, you can take your graph and squish it down to the Y axis. So notice for our graph, we've squished it down to the Y and this tells you the range. Now there are a couple different ways we can write the domain and range of a graph. One of the common things is to use interval notation and interval notation represents the domain and range using these brackets and parentheses. And you can actually see this in the example we have above. So notice that it looks like our domain goes from a value of negative four to positive five on the X axis. So this is what we said the domain was and for our range, it looks like we go from about negative one to positive two on the Y axis. So that's using interval notation. Now, there's another notation you can use called set builder notation and set builder notation uses these inequality symbols as opposed to the brackets and parentheses. Now something you need to keep in mind whenever using either one of these notations is that whenever you see these brackets or inequality symbols with the bar underneath, that means that you want to include whatever value you're looking at. So let's say we have this graph. For example, the values that you want to include are either closed dots or solid lines or curves. These are values that you always want to include in your domain and range. Now, when you see these parentheses or inequality symbols, without the bar underneath this means that you do not want to include those values and values that you do not want to include are situations where you see a hole in your graph. So anytime you see this hole or this open circle, that means you do not want to include it. Whereas any time you see a solid dot or a solid line or curve, you do want to include it. So given all this information, let's see if we can solve an example. So here we're asked to determine the domain and range of the graph below and to express our answer using interval notation. Now, what I'm first going to do is see if I can find the domain of this graph. And by the way, this is the same graph as you can see for both these diagrams. But for finding the domain recall that what we want to do is take our graph and squish it down to the X axis. So going down here, if we take this graph and we imagine squishing every single point down to the X axis, we're going to get a graph that looks something like this. We'll have a line that goes from negative four all the way to an X value of zero. And then we'll have another line that goes from positive one all the way to four. Now notice how there's an open circle here. So we have to include the open circle there as well. And then everywhere where else we have a closed circle because these circles were all closed. So looking at this squished graph, our domain is going to go from negative 4 to 0 on our x axis. And we need to include both these values since we have two closed circles. Now I can see that another one of these lines is going to go from positive one because this is an X value of one right here to four. So we're going to have one to four, we're going to include this positive one since we have a closed circle there. But our open circle is going to cause us to draw parenthesis here because we do not want to include that value. Now notice how we have multiple intervals here. Whenever you have multiple intervals or a jump in your graph, you need to use something called a union symbol which looks like this. So since we have multiple intervals, we need to put a union symbol between these to join the two intervals together. So this right here is the domain of our graph. Now, let's see how we can find the range and recall when finding the range, we need to take our graph and squish it down to the Y axis. So if I take this other graph we have drawn over here and I squish all these values down to the Y what we're first going to end up with is a line that goes from 1 to 3 on the Y axis. And then we're going to end up with another line that goes from negative three all the way to negative one. And we'll have an open circle at negative one since we have the open circle right there. So our range is going to go from negative three to negative one. And keep in mind we'll include the negative three, but we do not include the negative one since we have an open circle and our range will also go from positive 1 to 3. And we need to include both of these values since we have closed dots or I should say we have a solid liner curve. But still you want to include that value and we'll put the union symbol and then this is the range of our graph. So that is how you can find the domain and range of a function based on its graph. Let me know if you have any questions. And thanks for watching.

9

Problem

Problem

Find the domain and range of the following graph (write your answer using interval notation).

Hi, everyone. Welcome back. In the last video, we talked about how to find the domain and range of a function based on its graph. And in this video, we're gonna be looking at how you can find the domain of an equation. Now, when finding the domain of an equation, this is something that is a bit more complicated than with a graph. But we are gonna go over some of the common cases that you'll see that will hopefully make problems a lot more clear when you encounter them. So let's get into this. Now, recall when finding the domain of a function, you're looking for the allowed X values that you can have, but there are going to be certain situations where you'll have restrictions on these X values. So when finding the domain of an equation, you need to first identify the values that will break the function, any values that break your are going to be restrictions that you'll have and you need to be able to recognize these. And there are two common situations where we'll have restrictions. One of the common ones is whenever you have an X inside of a square root So the domain for X values inside a square root is going to be anything that does not make the inside of the square root negative. So you do not want to see a negative number underneath a square root. So let's take a look at this example. We're asked to find the domain of the function F of X is equal to the square root of X without graphing and to express our answer using interval notation. Now, remember you do not want the inside of the square root to be negative. So what that means is that our X values cannot be below zero. So any X values that are below zero will make this a negative right? We can have the square root of one that's just equal to one. We can even have the square root of zero, that's just equal to zero. But if we have the square root of negative one, well, you can try plugging that into your calculator, you're going to get an error. This is not something you're allowed to have. So your X values cannot be below zero. Therefore your domain is going to be from zero all the way to positive infinity. Now your domain can equal zero, that's perfectly fine, but it just cannot be below zero. So this would be the domain of the function. And this is the answer for this example. Now if we actually take a look at the graph of the square root of X notice that for the graph, this actually makes sense because notice that all of these values where X is above zero, these are all defined. But as soon as we look at the values where X is below zero, these values are not included. So it makes sense that our graph would be from zero all the way to positive infinity. Now remember this is one of the two common situations that you'll see where the domain has restrictions. The other common scenario is whenever you have X in the denominator of a fraction, because you need to make sure that for the domain that your X values do not make the denominator equal to zero, you cannot divide by zero in a fraction. So this is another situation you'll commonly run into. So let's take a look at this example. This example says given the function F of X is equal to two over X minus five, find the domain using interval notation. Now to find the domain of this function, the denominator cannot be equal to zero. So I'm just gonna write out that X minus five cannot equal zero. And I can just solve this like a mini equation. So we'll take five and we'll add it to both sides. That'll get the fives to cancel, giving me the X cannot be equal to five. So that means the X values that make the denominator zero are five. So our domain can equal five. So that means that our domain is going to be every value from negative infinity all the way up to five, but not including the five because we can't actually equal this number. And then we're also going to have an interval that goes from five all the way to positive infinity. So this basically says that we can have any real number except for the number five. And that is the domain of the function. So that is how you can find the domain of a function if you're given an equation rather than a graph. Hopefully you found this helpful and let me know if you have any questions.

11

Problem

Problem

Find the domain of $f\left(x\right)=\sqrt{x+4}$ . Express your answer using interval notation.

A

Dom: $\left[4,\infty\right)$

B

Dom: $\left[-4,2\right]$

C

Dom: $\left[2,4\right]$

D

Dom: $\left[-4,\infty\right)$

12

Problem

Problem

Find the domain of $f\left(x\right)=\frac{1}{x^2-5x+6}$ . Express your answer using interval notation.