WEBVTT
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We already know a little bit about functions.
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Let's talk about a special type of function called a one-to-one function.
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First of all let us remember that a function is a relation in which each x
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is associated with one and only one y.
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A one-to-one function is a function, first, in which each y
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is associated with one and only one x.
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It turns out that one-to-one functions are a special group of functions
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that are going to be very important to us.
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A one-to-one function has an inverse that is itself a function
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which as you will see is very important.
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When studying functions, we learned about something called the vertical line test
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and we can use that to quickly look at a graph of a relation, and decide if it is a function or not.
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Well to test if something is a one-to-one function,
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we have something called the horizontal line test.
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I bet that you know how that is going to work, so let's try it.
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I want to know if this is a one-to-one function,
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so I'm going to use the vertical line test to first determine is it a function.
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The vertical line test says if every vertical line you draw
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crosses the graph at most once, then the graph is a graph of a function.
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That happens here, so we at least know we have a function.
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Now we want to use the horizontal line test to tell if my function is one-to-one.
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So I start drawing, or at least imagining, horizontal lines
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that will cross the graph at most once, so this function is also one-to-one.
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Let's try another one.
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In determining if this is a one-to-one function,
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the first thing I need to know is, is it a function?
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Use the vertical line test to decide if it's a function.
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So every vertical line that I draw will cross the graph at most once. It's a function.
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Is it one-to-one? No because any horizontal line I draw that crosses the graph
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will cross it once or twice.
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Alright, let's draw an example. That one crosses it twice, so it's not one-to-one.
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Let's try one more. We want to know if this is a one-to-one function.
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First of all, is it a function?
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The answer is no because as soon as I draw a vertical line, it crosses the graph.
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I see it crosses here more than once.
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This one is not even a function,
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and we don't need to worry about the horizontal line test because it's not a function
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so it can't be a one-to-one function.
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Let's talk now about inverse functions.
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Suppose f is a one-to-one function.
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Then there is a function g that can be used to represent the inverse of f so that
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the composition of f with g is x
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and the composition of g with f at x is x.
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In addition, the domain of f will have to equal to the range of g and the range of f
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will equal the domain of g.
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We can use this to verify that two given functions are inverses of each other, and let's try that.
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We are asked to verify that these two functions,
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the cube root of x minus 2 for f of x,
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and g of x equals x cubed plus 2
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are in fact inverses of each other
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and we are to use composition of functions.
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So what we need to do is find f composed with g of x.
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Using the definition of composition,
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this is f of g of x; g of x is x cubed plus 2
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and we want to find f of x cubed plus 2.
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So we are going to substitute that here and work it out.
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So replace the x with x cubed plus 2,
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and then there is a minus 2. We'll work this out.
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2 minus 2 is zero, so this is the cubed root of x cubed, which will be x.
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That's good. That's the first step.
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The composition of f with g at x should result is just x, if they are inverses.
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Now let's try the other direction.
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So this time we want g of f of x.
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f of x is the cubed root of x minus 2. We want g of that.
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So we are going to replace x with this expression.
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So if we take the cube root of this and then cube it,
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we should come up with x minus 2
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and then we're adding 2, x minus 2 plus 2 is x.
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So using composition, we see that no matter the order of the g and the f,
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when we do the composition, we end up with x
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and this verifies that these functions are, in fact, inverses of each other.
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Let's look at an example that uses what we know
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about the graphs of a function and its inverse.
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For one thing, we know that the domain of a function
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should equal the range of its inverse,
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and the range of a function should equal the domain of the inverse.
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This essentially means that the x's and y's switch roles.
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So we can use that to help us to start to draw the graph of the inverse of f.
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Let's look at this point. This point has coordinates (0,1),
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and if we switch the x's and the y's,
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we are going to get a new point down here at (1,0).
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This point has coordinates (1,3).
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We switch the x and the y, we get a new point at (3,1).
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These are connected. It appears to be by a line segment,
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so let's connect them with a straight line segment.
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Now let's look at this one. This is the point (-2,0),
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and if we switch the coordinates we get a point at (0,-2).
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Again, it appears that they are connected by a straight line segment.
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Let's try this one. This is the point (-3,-2).
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Switch the x and the y and get a point that's at (-2,-3).
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That's all of the points that I have labeled
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that I can just directly switch the coordinates,
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but I still have to worry about this little piece of the graph up there.
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Does it go off like this? Or does it go up more?
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Well there is one more fact that we know that we can use.
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The graphs of a function and its inverse are symmetric
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with respect to the line y equals x. Let's draw that on our graph.
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The line y equals x is pretty easy to graph.
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It's going to contain points like (0,0), (1,1), (2,2),
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and going the other direction, (-1,-1), (-2,-2), and so on.
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Knowing that the graph of the function f and its inverse
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have to be symmetric with respect to this line tells me exactly how to finish my graph.
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I need this piece of the graph to be a mirror image of this piece on the top of f
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with respect to the line y equals x.
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So I want it to curve in this direction so that it's a mirror image,
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and this is my function f inverse.
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The inverse of a function f is a very important function and we give it a special name.
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This is the notation that you will use.
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f inverse, so it looks like f to the minus 1 power,
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but be careful because it doesn't mean f to the minus 1 power.
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It doesn't mean to do this. It is merely a notation for the inverse of the function f.
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Again, make a note of this and star it twice.
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f inverse of x does not mean to write 1 over the function.
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One more important fact for you to know is
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the graphs of a function and its inverse are symmetric
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with respect to the line y equals x.
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Now let's look at an example where we find the inverse of a function.
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So we have a linear function which is one-to-one,
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and we want to find its inverse and then state
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the domain and range of both of the functions.
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Let's start by writing down our function f of x equals 2x minus 4.
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We're going to start by switching f of x to y,
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and then because the domain of a function equals the range of its inverse,
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and the range of a function will be the same as the domain of its inverse,
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then the x's and the y's switch. So that's what we need to do, switch the x and the y.
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The next step in our process is to solve what we have for y.
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Let's add 4 to both sides, and then divide both sides by 2.
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This is the inverse and we want to rename it
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and call it f inverse of x equals x plus 4 over 2.
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Notice that the inverse function we found is in fact another linear function,
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and a more common way to write this would be 1/2 x plus 2.
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Finally, we are asked to state the domain and range of both of these functions.
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In the case of two linear functions, this is as easy as you can imagine
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because both of these linear functions have domain and range of all real numbers.
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So we simply just need to write that down.
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So this should be f inverse, and the domain of f inverse is still all real numbers
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and the range of f inverse.
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Let's look at an example with some nonlinear functions.
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Our next example asks us to find the inverse
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of f of x equals x squared plus 2 and state its domain and range.
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When you first look at this, it's a quadratic function and
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you should be asking yourself
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well wait a minute, there's something wrong here
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because when we look at the graph of x squared plus 2,
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it isn't one-to-one. Does f have an inverse? Can we even find it?
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Yes we can, but we have to pay attention to this inequality.
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It says use this function, but only for values of x greater than or equal to zero.
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So it's asking me to chop off part of the parabola, essentially.
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So that what I'm left with is a piece of a parabola that is one-to-one
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and then I can continue to find its inverse.
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This is an example of something called restricting the domain,
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and you will see it a lot in future math courses.
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So let's write down the function that we are trying to find an inverse for.
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We're going to change the f of x to a y.
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At this point, we switch the x's and the y's.
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Even this one become y is greater than or equal to zero.
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Now we can solve for y. Subtract 2 from both sides.
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We want to take the square root of both sides,
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and since this tells me that y should be greater than or equal to zero,
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I want the positive square root over here.
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So f inverse of x is the square root of x minus 2.
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Now we need to find the domain and range for each of the functions.
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Some of the work is already done for us.
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Since we restricted the domain of f to be values greater than or equal to zero, there is our domain.
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The domain of f is values of x greater than or equal to zero,
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so we will write it in interval notation.
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The range of f you can get several different ways,
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but since I've already drawn the graph up there, I think I'd like to use the graph.
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The range of f is the values that y can have.
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This was x squared plus 2, so the smallest y value is 2,
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and it can be anything bigger. I'm halfway there.
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I know that the domain of f is the same as the range of f inverse
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So the range of f inverse must be zero to infinity, including the zero.
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I also know that the domain of f inverse
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is the same as the range of the original function
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so that would be from 2 to infinity.
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In this example, we'll look at finding the inverse of a rational function.
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So here, f of x is 1 over x minus 3. Replace f of x with y.
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Switch the x and the y because you are looking for the inverse.
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And then we are going to solve for y. What can I do?
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Multiply both sides by y minus 3 is a good idea.
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Keep in mind your goal is to get this y by itself,
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so let's go ahead and divide by x on both sides.
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Now we can add 3 on both sides.
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So the inverse is 1 over x plus 3, another rational function.
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The last thing I need to do is find the domain and range for each of the two functions.
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Let's start with f. When I look at this, I see right away that x can't equal 3
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because that would give me zero in the denominator.
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So the domain of f is all real numbers except 3.
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To get the range of f, I'll need to look at its graph.
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We can use transformations of our basic rational function 1 over x.
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We're going to take the graph of 1 over x and then shift it to the right 3 units.
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So the first thing that will do is move the vertical asymptote to the right 3 units.
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It's not going to move this up or down at all
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so we just need to sketch in the proper shape with a couple of points labeled.
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This confirms to me that x can't be 3, which is what we said,
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and it also shows me that y is never equal to zero.
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So the range of f is all real numbers except zero.
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That takes care of f. What about the inverse?
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Again, we can see by looking at the inverse function that x can't be zero.
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So the domain is all real numbers except zero.
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I can't easily see the range of this looking at the equation,
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so we'll need to draw the graph of the inverse function and find the range from that.
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This is the graph of 1 over x, moved up three units.
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So this time it moves the horizontal asymptote up three units,
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but it doesn't shift it left or right.
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Again, we can confirm that x can't be zero, so our domain looks good.
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Now we'll find the range. The horizontal asymptote is at y equals 3;
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y is never equal to 3 on the graph.
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So this is all real numbers except 3.
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You'll need to practice these skills dealing with inverse functions and one-to-one functions.
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These concepts will show up many times in your mathematical studies.