WEBVTT
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In this video, we're going to learn about functions. We'll begin our discussion
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with relations, domain, and range. In a previous section, we learned that we can write an equation
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to discuss the relationship between two variables. For example in a square,
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if we let the length of one of the sides of a square equal x and the area of that square
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equal y, then I can write an equation that demonstrates the relationship of those variables.
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The area of the square y is equal to the length of the sides squared,
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or x squared. So this equation y equals x squared shows a relationship between these two variables.
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A relation is a set of ordered pairs. A set of ordered pairs is called a relation.
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The set of all of the x-coordinates is the domain of that relation, and the set of the
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y-coordinates is called the range of a relation. Now equations such as this y equals x squared
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are also called relations since equations in two variables define a set of ordered pairs.
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A function is a special type of relation. A function is a set of ordered pairs that assigns to each x-value exactly
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one y-value. In just a moment, we're going to explore this definition a little further.
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In these two examples, we are going to state the domain and range of the given relation.
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And then we're going to determine if the relation is a function. Recall the definition of a function.
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A function is a set of ordered pairs that assigns to each x-value exactly one y-value. Let's take a look at these relations.
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Here's our set of ordered pairs. The domain is the set of the x-values of these ordered pairs.
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So in this case the domain is the set of numbers that contains four, negative three,
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and nine. Many times it's convenient to write these in ascending order,
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but the set of numbers is still the same regardless of the order that this is in.
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In any case, we do not duplicate a number when we write it in a set. So I will write this set
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as negative three, four, and nine. The range is the set of y-values in these ordered pairs.
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The y-values here are negative four. Here's another negative four. Remember
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we'll only write this once. And zero. So the range is the set negative four and zero.
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These are the domain and range. The next part of the instructions asks us to determine whether or
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not this relation is a function. So we need to determine if each element in the domain
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is paired with exactly one element in the range. Negative three is paired only with negative four.
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Four is paired only with negative four. And nine is paired only with zero.
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So even though the y-value negative four is repeated, each x-value is assigned to
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exactly one y-value. Therefore, this relation is a function. So we will say yes it is a function.
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Here's our next relation. Let's look at the domain. The domain is the set of x-values
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in our relation. We have the x-values zero, three, negative two, and zero
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is repeated. If I write these in ascending order, my domain is negative two, zero, and three.
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Remember, we only write the zero once, even though it shows up in two ordered pairs.
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The range is the set of the y-values in the relation. The y-values include the numbers five,
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zero, ten, and negative eight. Writing these in ascending order, we have negative eight,
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zero, five, and ten. Now let's determine whether or not this relation is also a function.
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This element in the domain negative two is paired only with ten.
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The element zero, however, is paired both with five and negative eight. Since this element in
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the domain is paired with more than one element in the range, this relation is not a function.
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Relations and functions can be described by the graph of the relation.
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This picture shows a relation because it has four ordered pairs identified, and recall that
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a relation is a set of ordered pairs. Is the graph shown the graph of a function?
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Let's look a little more closely at these ordered pairs. We'll write down the ordered pairs in this relation.
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The first ordered pair on the left, far left, side has the ordered pair negative three, negative two.
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This ordered pair is negative two, positive two. This ordered pair is negative two, negative three. And this
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final ordered pair is one, one. Recall that to be a function each x-value must be paired with exactly one y-value.
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Let's look at the domain of this relation. It includes the numbers negative three,
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negative two, and one. Notice that negative two is paired with both two and negative three.
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Because it is paired with more than one y-value, this is not a function.
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Let's also look at the graph to see what we can tell about this relation and not being a function
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as we look at the ordered pairs here. These two ordered pairs both share the same x-value. The x-coordinate
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is negative two. It is paired with both positive two and negative three. Thus again, this graph this relation is not a function.
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You may have noticed in the last example that when a vertical line can be drawn on the graph and it
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intersects the graph at more than one point, we know then that the relation is not a function
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because the ordered pairs that fall on that vertical line have the same x-value
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that are paired with more than one y-value. We can use this trait to learn about whether or
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not graphs are functions. This is called the Vertical Line Test. If a vertical line can be
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drawn so that it intersects a graph more than once, the graph is not the graph of a function.
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Let's take a look at this example. If I draw a vertical line through this graph
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anywhere on this graph, notice this vertical line will only intersect the graph
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one time. No matter where I draw this vertical line throughout this graph, the vertical line
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is only intersecting one time. Thus, every x-value here is paired with only one y-value.
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In this picture, if I draw a vertical line, here sure it only intersects one time with the graph.
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However, if I draw that vertical line here, notice that this vertical line
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intersects the graph more than one time. If a vertical line drawn anywhere on this graph
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intersects with this graph at more than one location, then the graph is not a function.
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So this graph is not the graph of a function. Whereas, the first graph,
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this one, where the vertical lines intersected only once, this is the graph of a function.
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Recall that the graph of a linear equation is a line. A line that is not
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vertical will always pass the Vertical Line Test. Thus, a non-vertical line is a function.
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An equation that is written in the form x equals c is a vertical line. And vertical lines
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clearly will not pass the Vertical Line Test, as all of the ordered pairs on that line share
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the same x-coordinate. Thus, an equation that is written in the form x equals c is not a function.
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Let's take a look at these two examples. Determine whether the equation describes a function.
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Our first equation is y minus five x equals three. We can recognize this as a linear equation because
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it can be written in the form A x plus B y equals C. In this equation, both A and B are not zero. It
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cannot be written in the form x equals c. It's not a vertical line. Thus, it is a function.
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In our second equation, we have x equals two. We should recognize this is in the form x equals c,
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which represents a vertical line. And again, vertical lines do not pass the Vertical Line Test.
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They have an infinite number of ordered pairs that have the same x-value. That x-value is
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paired to more than one y-value. Thus, this equation does not represent a function.
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When we first spoke of functions, we described a function as a relation such that every
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x-value in the relation corresponds to exactly one y-value in the relation. When that happens, that
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relation is a function. We also identified that linear equations that are not vertical represent
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functions because every x-value will relate only to one y-value. Thus, we extend our understanding
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of a function by saying that the variable y is a function of the variable x. We can substitute
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any value in the domain of the function for x. That would then give us a corresponding value of y.
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The variable x, thus, is the independent variable because any value in the domain can be assigned
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to x. The variable y is the dependent variable because its value depends on the value of x.
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We can also examine this function with a different notation. To identify that this
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equation represents a function, we can write this as f of x equals two x plus one.
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This is known as function notation. Note that the way we say this is f of x. This is not f times x. What this
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means is that the value of the function, which is y, depends upon the value of x.
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If I write f of one, in this way, this means the value of the function when x is equal to one.
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I can find f of one by substituting one for the x-value in the equation. So f of one
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would equal two times one plus one. Two times one plus one is three. So f of one is three.
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This can also be written as an ordered pair for the function. The x-value is one,
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and the y-value is three. Common letters used to represent functions include f,
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g, and h. So you may see f of x, g of x, h of x. Quite frankly, many letters,
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any letter, can be used to represent a function. These are just the most common.
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Consider the function f of x equals x squared plus three. Let's find these three values beginning
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with f of negative four. To find f of negative four, we will substitute negative four into
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the x-value of the function, giving us negative four squared plus three. Negative four squared
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is positive sixteen, and then we are adding three, giving us nineteen. So f of negative four
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is nineteen. Recall, I can write this as an ordered pair as well. This notation means that the ordered pair
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negative four, nineteen is on the graph of this function. I do want to point out the various uses of parentheses here.
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Notice that here f of negative four. Remember, we talked about the fact that this does not mean f times negative four.
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It means the value of the function when x is negative four. This set of parentheses here means that we are
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multiplying negative four times itself two times. This is a multiplication because of the exponent.
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And then this set of parentheses is used to represent an ordered pair negative four, nineteen.
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They are all parentheses, but yet they have specific uses and meanings in these problems.
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Let's look at the next example. We're to find f of zero. We will substitute zero for the x in the
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function. Giving us zero squared plus three. Zero times zero is zero, and then plus three
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gives us three. So f of zero is equal to three. This is the value of the function when x is zero.
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It can be listed as an ordered pair on the graph of this function. The ordered pair is zero, three.
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And our last problem is f of five. Again, we will substitute five for the x in the function,
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giving us five squared plus three. Five squared is five times five or twenty-five,
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and then we are adding three, giving us twenty-eight. So f of five is twenty-eight.
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And again, this can be written as an ordered pair on the graph of the function five, twenty-eight.