Hi and welcome. My name is Anthony Varela. And today, I'm going to introduce functions. So first, we're going to talk about the difference between functions and relations. Then we'll see how to recognize a function from a table of values and from a graph. And finally, we'll talk about function notations, so how do we write functions, and how do we read them.
So first, let's talk about relations. Well, the relation between two sets matches elements of one set with elements of the other set. So there's a relationship between sets of data. So for example, if I'm collecting information about how many siblings and how many pets people have, I might collect my data and represent them as coordinate pairs.
So 0, 2, would be one set of data from one individual that tells me they have zero siblings and two pets. 1 comma 2 is another set of information that lets me know that they have one sibling and two pets. And then 2, 0, would indicate one person has two siblings but zero pets.
Now, I could represent this in a table and use an x column and a y column. So here, I just have my x values 0, 1, 2, and then my y values 2, 2, and 0. And we can give these columns special names. We can say that the x column represents the domain. And the y column represents the range.
So what are the domain and the range? Because we use this language a lot, particularly with functions. So the domain is the set of all input values of a function or relation. The range is the set of all output values of a function or relation. So there's a relationship then between elements in the domain and elements in the range that one corresponds to an element in the other set.
So how does this differ from functions? Well, a function is a special type of relation. And it is a relation in which every element in the domain corresponds to exactly one element in the range. And this is our important clause here-- exactly one element in the range.
So we're going to take a look at a couple of tables of values and determine if that represents a function. And we're going to be looking to make sure that every element in the domain corresponds to only one element in the range, no more than one.
So here, taking a look at this first table, here are all the elements in our domain. Here are all of the elements in our range. And the blue arrows represents the correspondence between the elements. So 0 corresponds to 2. 1 corresponds to 9. 2 corresponds to 7. 3 corresponds to negative 3. And 4 corresponds to 5.
Now, the first thing we should do is make sure that this represents a relation. And it certainly does. There is a line that connects every element in the domain to an element in the range. So this is a relation. But is it a function? And we see that there are no more than one blue arrows branching out from elements in our domain. So we know that this is indeed a function.
Now, let's take a look at the second table. And this one can be tricky. Now, looking at the blue arrows, we can see that both 0 and 1 correspond to 2 in the range. And so we might be tempted to say this does not represent a function. But this is actually OK.
We can have more than one element in the domain corresponding to the same element in the range. We just can't have it be the other way around. So it's OK to have more than one arrow pointing to any element here. But we cannot have more than one arrow branching out from any particular element in the domain.
So that would be illustrated in this table here. This does not represent a function. Because if you look at this point three, we see that it corresponds to nine, and it corresponds to five. So this element, which is in the domain, corresponds to two elements in the range. So that violates the rule for what a function can be. So these two tables represent functions. This one does not.
How can we recognize a function on a graph? Well, we use something called the vertical line test. And so what we do with the vertical line test is we have a graph of a function. And we take a vertical line. So this could be a piece of paper, a straight edge, or a ruler. And you just swipe it across the graph of the function.
And what you're looking for is you're looking to make sure that the graph only touches or intersects this vertical line at only one point at any given time. So in other words, you're checking graphically to make sure that every element in the domain corresponds to only one element in the range. And so I notice here that this does represent a function because the red line intersects the blue line at only one point across the entire graph.
So maybe it's easier to see a nonexample. So this graph is not a function. It fails the vertical line test. Because we see right here, the graph touches the vertical line at only one point. But as we continue to swipe this vertical line, we see that there's one, two points of intersection.
And if there's ever more than one point of intersection, it fails the vertical line test. So this is not a function. So graphically, we use the vertical line test as a tool to show if the relation is a function.
Lastly, we're going to talk about function notation. So we're going to be comparing functions to equations that we're used to seeing. So if we had this equation y equals 3x minus 5, well, y is actually a function. It passes the vertical line test if you were to graph this. In fact, I know all linear lines except for vertical lines would pass the vertical line test.
So we can represent this as a function. So how we can show that it's a function is we use f of x equals 3x minus 5. So we use f and x in parentheses. This does not mean f times x. We read this f of x equals 3x minus 5. So in function notation, we see f of x. And that's exactly how we read it-- f of x.
Now, with equations, you might be asked to solve an equation so solve y equals 3x minus 5 when x equals 3. With functions, we're asked to evaluate a function at a given point. So given f of x equals 3x minus 5, evaluate f of 3. So this is another weird thing in function notation. This means what is the value of the function when x equals 3?
So solving the equation, what we would do is just substitute 3 in for x. Solve for y. When we're evaluating our function, we say that f of 3 equals 3 times 3 minus 5. So we're still substituting this value in for our variable that's in our function. But we just write it a bit differently-- f of 3.
So in function notation, f of x can be thought of as y in equations-- y equals 3x minus 5, f of x equals 3x minus 5. But we evaluate a function, we can say f of a was evaluating the function f of x when x equals a. So some subtle differences between equations and functions but pretty similar ideas.
So let's review our introduction to functions. Well, we talked about a function being a relation in which every element in the domain corresponds to exactly one element in the range. So remember, the domain is the set of all those input values. The range is the set of all those output values. And we're looking to make sure that every element in the domain corresponds to only one element in the range.
We talked about the vertical line test being a graphical tool to show if a relation is a function. So if our curve on our graph touches that vertical line only once as you swipe that line across the entire function, that means it validates that it is a function.
And then we talked about function notation f of x. So not f times x, but f of x. And we evaluate functions, we say f of a equals f of x when x equals a. So thanks for watching this introduction to functions. Hope to see you next time.