WEBVTT mathematics/algebra-2/eaton
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Welcome to Educator.com.
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For today's Algebra II lesson, we are going to be discussing relations and functions.
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And recall that some of these concepts were discussed in Algebra I, so this is a review.
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And if you need a more detailed review, check out the Algebra I lectures here at Educator.
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Beginning with the concept of the **coordinate plane**: the coordinate plane describes each point as an ordered pair of numbers (x,y).
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The first number is the x-coordinate, and the second is the y-coordinate.
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For example, consider the ordered pair (-4,-2): this is describing a point on the coordinate plane with an x-coordinate of -4 and a y-coordinate of -2.
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Or the pair (0,2): the x-value would be 0, and the y-value would be 2; this is the point (0,2) on the coordinate plane.
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Or (3,5): x is 3; y is 5.
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Also, recall that the quadrants are labeled with the Roman numerals: I, and then (going counterclockwise) quadrant II, quadrant III, and quadrant IV.
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In the coordinate pairs in the first quadrant, the x is positive, as is the y.
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In the second quadrant, you will have a negative value for x and a positive value for y, such as (-2,4)--that would be an example.
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In the third quadrant, both x and y are negative; and then, in the fourth quadrant, x is positive; y is negative.
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And we will be using the coordinate plane frequently in these lessons, in order to graph various equations.
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Recall that a relation is a set of ordered pairs.
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The domain of the relation is the set of all the first coordinates, and the range is the set of all the second coordinates.
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A relation is often written as a set of ordered pairs, using braces to denote that this is a set,
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and then the ordered pairs, each in parentheses, separated by a comma.
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Sometimes, the relation is represented as a table; so, -2, 1; -1, 0; 0, 1; and 1, 2.
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We will be doing some graphing of relations also, in just a little bit.
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So, as discussed up here, the domain is the set of all first coordinates.
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And the set of all first coordinates here would be {-2, -1, 0, 1}.
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The range is the set of the second coordinates; so the range right here--all the y-values--is {1, 0, 1, 2}.
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However, you don't actually need to write the 1 twice; so in actuality, it would be written as such.
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It is OK to repeat the values if you want, but usually, we just write each value in the domain or range once; each is represented once.
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Functions are a certain type of relation: so, all functions are relations, but not all relations are functions.
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A function is a relation in which each element of the domain is paired with exactly one element of the range.
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For example, consider the relation shown: {(1,4), (2,5), (3,8), (4,10)}.
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Each member of the domain corresponds to exactly one element of the range.
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We don't have a situation where it is saying {(1,4), (1,6), (1,8)}, where that member of the domain is paired with multiple members of the range.
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Another way, again, to represent this as a table--another method that can be used--is mapping.
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And mapping is a visual device that can help you to determine if you have a function or not,
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by showing how each element of the domain is paired with an element of the range.
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A map would look something like this: over here, I am going to put the elements of the domain, 1, 2, 3, and 8;
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over here, the elements of the range: 4, 5, 8, and 10.
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And then, using arrows, I am going to show the relationship between the two.
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So, 1 corresponds to 4 (or is paired with 4); 2 to 5; 3 to 8; and (this should actually be 4) 4 to 10.
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OK, so as you can see, there is only one arrow going from each element of the domain to each element of the range.
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And that tells me that I do have a function.
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Let's look at a different situation, using a table form: let's look at a second relation.
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In this one, I am going to have (-2,2), (-3,2), (-4,5), and (-6,7).
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And I am going to go ahead and map this: -2, -3, -4, and -6: these are my elements of the domain.
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For the range, I don't have to write 2 twice; I am just going to write it once; 5, and 7.
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OK, -2 corresponds to 2; -3 also corresponds to 2; -4 corresponds to 5; and -6 corresponds to 7.
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This is also a function, so both of these are relations, and they are also functions.
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It is OK for two elements of the domain to be paired with the same element of the range; this is allowed.
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What is not allowed is if I were to have a situation where I had {1, 2, 3}, {4, 5, 6}; and I had 1 paired with 4, and 1 paired with 5.
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So, if you have two arrows coming off an element of the domain, then this is not a function.
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Here are two examples of relations that are also functions.
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There is a specific type of function that is called a one-to-one function.
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And a function is one-to-one if distinct elements of the domain are paired with distinct elements of the range.
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In the previous example, we saw a situation where we did have a one-to-one function, and another situation where we did not.
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OK, so to review: the ordered pairs in that first function that we just discussed were (1,4), (2,5), (3,8), and (4,10).
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OK, and we can use mapping, again, to determine what the situation is with this relation (which is also a function).
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The domain is {1, 2, 3, 4}; and the range is {4, 5, 8, 10}.
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When I put my arrows to show this relationship, you see that distinct elements of the domain are paired with distinct elements in the range.
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1 is paired with 4; they are each unique--each pair is unique.
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Looking at the other function that we discussed: the pairs are (-2,3), (-3,2), (-4,3)...slightly different, but the same general concept...slightly different, though.
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OK, here I have -2, -3, -4, and -6; over here, in the range, I have 3, 2...I am not going to repeat the 3--I already have that...and then 7.
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-2 corresponds to 3; -3 corresponds to 2; -4 also corresponds to 3; -6 corresponds to 7.
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This is still a function; OK, so these are both functions: function, function.
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However, this is a one-to-one function; this is not one-to-one.
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They are both functions, since each element of the domain is paired only with one element of the range.
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But in this case, it is not a unique element of the range: these two, -2 and -4, actually share an element of the range.
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In other words, this is unique; it is a one-to-one correspondence.
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OK, we can graph relations and functions by plotting the ordered pairs as points in the coordinate plane, as discussed a little while ago.
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There are a couple of types of graphs that you can end up with.
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The first is discreet, and the second is continuous; let's look at those two different types.
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Consider this relation: OK, so if I am asked to graph this relation, I am going to graph each point:
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(-4,-2): that is going to be right here; (-2,1)--right here; here, (0,2), 0 on the x, 2 on the y.
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This is a discrete function--discrete graph--discrete relation.
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This actually is both a relation and a function; so it is a discrete relation or a discrete function.
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And the reason is because I have a set of discrete points; they are not connected.
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And I can't connect them, because I haven't been given anything in between, or a way to know if or what lies in between these.
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I can't just connect them when I don't know; there could be a point up here, or actually this is just the entire relation.
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So, I can just work with what is given.
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OK, a different scenario would be if I am given a relation y=x+1.
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And I can go ahead and plot this out, if I say, "OK, when x is -1, -1+1 is 0; when x is 0, y is 1; when x is 1, 1+1 is 2; when x is 2, y is 3."
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OK, so I am going to go ahead and plot this out.
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When x is -1, y is 0; when x is 0, y is 1; when x is 1, y is 2.
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Let's remove this out of the way.
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When x is 1, y is 2; when x is 2, y is 3.
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Now, I have a set of points, because these are the points I chose.
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But because I am given this equation, there is an infinite number of points in between.
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I could have chosen an x of .5 to get the value 1.5 here, to fill that in--and on and on, until this becomes continuous and forms a line.
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So, this is a continuous function: the graph is a connected set of points,
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so the relation (or the function in this case, since we do have a function) is continuous relation or continuous function,
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because I have a line; whereas this, which is a set of points, is a discrete relation.
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OK, one visual way to tell if a relation is a function is using the vertical line test.
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And a relation is a function if and only if no vertical line intersects its graph at more than one point.
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This is most easily understood through just working through an example.
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Consider if you were given the following graph.
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OK, the vertical line test: what you are seeing is, "Can you put a vertical line
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somewhere on the graph so that it intersects the graph at more than one point?"
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And I can: I put a vertical line here, and it intersects this graph at 1, 2, 3 places.
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Over here, it only intersects at one place; that is fine; but if I can draw a vertical line *anywhere* on the graph
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that intersects at more than one place, then we say that this failed the vertical line test.
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And when something fails the vertical line test, it means that it is not a function.
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The reason this works is that, if two or three or more points share the same x-value,
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then they are going to lie directly above or below each other on the coordinate plane.
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For example, looking right here, I have x = 3; x is 3; y is 0.
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Then, I look right above it, up here: again, x is 3, and y is...say 2.1--pretty close.
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Then, I look up here; again, x is 3, and y is about 4.6, approximately.
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So, when x-values are the same, but then the y-values are different, that is telling me
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that members of the domain are paired with more than one member of the range; by definition, that is not a function.
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OK, consider a different graph--consider a graph like this of a line--a straight line.
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OK, now, anywhere that I pass a vertical line through--anywhere on this graph--it is only going to intersect at one point.
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So, this passed the vertical line test.
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Therefore, this line, this graph, represents a function.
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OK, so the vertical line test is a visual way of determining if a relation is a function.
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Working with equations: an equation can represent either a relation or a function.
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If an equation represents a function, then there is some terminology we use.
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And let's start out by just looking at an equation that represents a function.
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The variable corresponding to the domain is called the independent variable, and the other variable is the dependent variable.
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So, here I have x and y; and let's look at some values--let's let x be -1.
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Well, -1 times 2 is -2, minus 1--that is going to give me -3.
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When x is 0, 0 times 2 is 0, minus 1 is -1.
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When x is 1, 1 times 2 is 2, minus 1--y is 1.
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When x is 2, 2 times 2 is 4, minus 1 gives me 3.
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So, looking at how this worked, x is the independent variable.
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The value of x is independent of y; I am just picking x's, and here it could be any real number.
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We sometimes also say that this is the input; and the reason is that I pick a value for x (say 0),
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and I put it in--I input it into the equation; then, I do my calculation, and out comes a y-value.
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So, the value of y is dependent on x; therefore, it is the dependent variable; and we also sometimes say that it is the output.
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You put x in and do the calculation; out comes the value of y; so x is independent, and y is dependent.
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The notation that you will see frequently in algebra is function notation.
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We have been writing functions like this: y = 4x + 3; but you will often see...
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instead of an equation written like this, if it is a function, you will see it written as such.
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And when we say this out loud, we pronounce it "f of x equals 4x plus 3."
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And we are talking about the value of a function for a particular value of x.
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So, we say, "The function of f at a particular x."
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Let's let x equal 3; then, we can talk about f of 3--the value of the function, the value of y,
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of the dependent variable, when the independent variable, x, is 3.
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And in that case, since it is telling us that x is 3, I am going to substitute in 3 wherever there is an x.
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And I could calculate that out to tell me that f(3) is...4 times 3 is 12, plus 3...so f(3) is 15.
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Here, x is an element of the domain, of the independent variable; f(x) is an element of the range.
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So again, we are going to be using this function notation throughout the remainder of the course.
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Looking at the first example: the relation R is given by this set of coordinate pairs.
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Give the domain and range, and determine of R is a function.
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Well, recall that the domain is comprised of the first element of each of these coordinate pairs.
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So in this case, the domain would be {1, 2, 6, 5, 7}.
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The range: the range is comprised of the second element of each ordered pairs, so I have 4, 3...4 again;
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I don't need to write that again; 3--I have 3 already; and 5; I am just writing down the unique elements.
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This is the domain, and this is the range.
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Now, is R a function? Well, I can always use mapping to just help me determine that.
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And I am going to write down my members of the domain, and my elements of the range.
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And then, I am going to use arrows to show the correspondence between each:
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1 and 4--1 is paired with 4; 2 is paired with 3; 6 is also paired with 4; 5 is paired with 3; and 7 is paired with 5.
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Now, I am looking, and I only see one arrow leading from each element of the domain.
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There is no element of the domain that is paired with two elements of the range.
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So, in this case, this is a function; so, is R a function? Yes, this relation is a function--R is a function.
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So, always double-check and make sure you have answered each part.
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I found the domain; I found the range; and I determined that R is a function.
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OK, the relation R is given by the equation y=2x²+4; is R a function?
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What are the domain and range? Is R discrete or continuous?
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Let's just look at some values for x and y to help us determine if this relation is a function.
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If I let x equal -1, -1 times -1 is 1, times 2 is 2, plus 4 is 6.
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OK, if x is 0, this is 0, plus 4--that gives me 4.
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If x is 3, 3 squared is 9, times 2 is 18, plus 4 is 22.
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So, as you are going along, you can see that, for any value of x, there is only one value of y; therefore, R is a function.
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What is the domain? Well, I could pick any real number for an x-value that I wanted, so the domain is all real numbers.
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You might, at first glance, say, "Oh, the range is all real numbers, as well"; but that is not correct, because look at what happens.
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Because this is x², whenever I have a negative number, it becomes positive; if I have a positive number, it stays positive, of course.
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Therefore, if I have, say, -1, that becomes 1; this becomes 6.
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So, I am not going to get any value lower than...for y, the smallest value I will get is for when x is 0.
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OK, so if x is 0, y is 4; because -1 is going to give me a bigger value--it is going to give me 6.
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If I do -2, that is going to be 4 times 2 is 8, plus 4 is 12.
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So, the lowest value that I will be able to get for y will occur when x is 0.
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And that is going to give me a y-value of 4.
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Therefore, the range is that y is greater than or equal to 4.
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So, the most difficult part of this was just realizing that the range is not as broad as it looked initially.
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Because this involves squaring a number, there is a limit on how low you are going to go with the y-value.
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So, this is a range with a domain of all real numbers, and a range of greater than or equal to 4.
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OK, in Example 3, graph the relation R given by 2x - 4y = 8.
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Is R a function? Find its domain and range. Is R discrete or continuous?
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OK, so graph the relation given by 2x - 4y = 8.
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Let's go ahead and find some x and y values, so that we can graph this.
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When x is 0, we need to be able to solve for y; when x is 0, let's figure out what y is.
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0 - 4y equals 8; therefore, y equals -2 (dividing both sides by -4).
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OK, when x is 2, 2 times 2 minus 4y equals 8; that is 4 minus 4y equals 8; that is -4y equals 4; y = -1.
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And let's do one more: when x is -2, this is going to give me -4 - 4y = 8.
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That is going to then give me, adding 4 to both sides, -4y = 12, or y = -3; that is good.
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All right, so when x is 0, y is -2; when x is 2, y is -1; when x is -2, y is -3, right here.
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I am asked to graph it; and I have some points here that I generated,
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but I also realize that I could have picked points in between these, which would actually end up connecting this as a line.
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So, I am not just given a set of ordered pairs; I am given an equation that could have an infinite number of values for x,
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which would allow me to graph this as a continuous line.
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Therefore, I graphed the relation...is R a function?
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Is R discrete or continuous? Well, I have already answered that--seeing the graph of this, I know that this is continuous.
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And let's see, the next step: is R a function?
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Yes, it is a function, because if I look, for every value of x (for every value of the domain), there is one value only of the range.
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So, every element of the domain is paired with only one element of the range.
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It is continuous, and it is a function.
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Find the domain and range: well, this is another case where I could choose x to be any real number, so it would be all real numbers--any real number.
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Here, the situation is the same for the range--all real numbers.
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Depending on my x-value, I could come up with infinite possibilities for what the range would be, what the y-value would be.
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R is a function; its domain and range are all real numbers; and this is a continuous function.
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OK, in Example 4, we are given f(x) = 3x² - 4, and asked to find f(2), f(6), and f(2k).
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First, f(2): recall that, when you are asked to find a function for a particular value of x,
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you simply substitute that value for x in the equation; so f(2) equals 3(4) - 4, so that is 12 - 4; so f(2) = 8.
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Next, I am asked to find f(6), and that is going to equal 3(6²) - 4.
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f(6) = 3(36) -4, and that turns out to be 108 - 4, so f(6) is 104.
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Now, at first, this f(2k) might look kind of difficult; but you treat it just the same as you did with the numbers, when x is a numerical value.
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Everywhere I see an x, I am going to insert 2k.
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And figuring this out, 2 times 2, 2 squared, is 4; k times k is k².
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3 times 4 is 12, so I have 12k² - 4; so f(2k) = 12k² - 4.
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So again, if you are asked to find the function of a particular value of x, you simply substitute whatever is given, including variables, for x.
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That concludes this lesson of Educator.com; I will see you back here soon!