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Introduction to Functions

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Today, we're going to talk about functions. A function is just a special type of relation. And a relation is just a relationship between two sets of values where each element in one set of values corresponds to an element in another set of values.

So we're going to look at some examples to determine whether something is or is not a function. So let's look at an example. Let's say I'm comparing two sets of data, age and weight. And I want to look at the relationship between these two sets of data.

I can look at three ordered pairs. And in each ordered pair, the first value, which is our age, is represented by a value in the domain of our relation. And the second value in each ordered pair, or the weight, is represented by a value in the range of this relation.

So I can also represent these values in the domain and the range as a table. And we represent the values in our domain with the x variable and we represent the values in our range with the y variable. So looking at our ordered pairs, creating a column of data with just the values in our domain we have 0.5, 1.0, and 2.0.

And we can create a column with just the values in our range. 16, 21, and 26. And we know that this data represents a relation because each element in the domain will map or corresponds to an element in the range.

So let's talk about what it means to be a function. A function is a special type of a relation where every element, x, in the domain corresponds to exactly one element in the range. And the opposite does not necessarily hold true. You can have an element, y, in the range that corresponds to more than one element in the domain.

So let's look at a couple of examples of relations represented by tables. And one of these is a function and one of them is not. So let's see. We'll look at the first relation that's represented by this table and we could say that the domain has the elements negative 1 and 0. We don't need to write negative 1 here twice.

The range has the elements 4, 6, and 8. So we can see by looking at our table that the value of negative 1 corresponds to 4. But the value of negative 1 also corresponds to 6. And the value of 0 corresponds to 8.

So here, we can see that we have an element in our domain, negative 1, that corresponds to more than one element in the range. So this is not a function.

Let's look at our second table. The elements in our domain are negative 1, 0 and 1. And the elements in our range is just 3.

So we can see that negative 1 corresponds to 3, 0 corresponds to 3, and 1 corresponds to 3. But we have each element in our domain corresponding to only one element in our range. They all only correspond to 3. So this is a function.

So now, let's look at how we can determine whether a relation is a function by looking at its graph. Remember that the definition of a function is that every element in the domain can correspond to only one element in the range. So every x value can have only one y value.

So on a graph, that means that at any value for x, the curve crosses a vertical line through that x value only one time. We call this the vertical line test. And what that looks like is that if I pick any value for x and draw a vertical line through that x value on the x-axis, then it should cross my graph, the curve of f of x, only one time.

So here we see that it crosses only one time. And that has to hold true for any value of x. So here again, I can see that my graph of f of x, the curve, crosses the vertical line only one time. So we can see that this graph of y equals x squared that this is a function.

So let's talk a little bit about function notation. I have an example of a linear equation, y equals 2x plus 7. I know that a line is also going to be a function because every line would pass the vertical line test. So I can write this equation for a line in function notation, which would look like this.

Function notation is represented with the f and then an x in parentheses. This could also be another variable besides x. And we say this as f of x. It does not mean f multiplied by x, but the way that we say it is f of x.

And you can see that we're replacing this f of x by the y that we had in our original equation. So when we're solving an equation, we might solve this equation, y equals 2x plus 7, when x is equal to 3. So an example of the problem would be to solve this equation when x is equal to 3.

In function notation, we don't solve it in the way that we would solve an equation, but we can evaluate an equation or an expression that's written in function notation. And to do that, we would say, for example, for this function, f of x is equal to 2x plus 7, find f of 3. And what we want to do when we're finding f of 3 is we're replacing or we're substituting 3 for every x that's in the expression and then we evaluate.

So let's go over our key points from today. The first value in the ordered pair, x, represents the elements in the domain, and the second value, y, represents elements in the range. A function is defined as a relation in which every element in the domain corresponds to exactly one element in the range.

If a graph represents a function, at any value for x, the curve of the function touches or crosses any vertical line only once. This is called the vertical line test. And function notation is used to evaluate a function for a given value in the domain of the function.

So I hope that these key points and examples helped you understand a little bit more about functions. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.