We’re now going to review the very important function concept. Now, somewhat in the abstract, we’re going to look at the terminology, especially as it relates to inverse functions. Now, this material is important in its own right, but it’s also going to be the key to define the other important class of functions we’re meeting this week– namely, the logarithmic functions. So a function, let’s give it the name phi, means a rule that gives to any point x in a set D, called the domain of the function, a value that we denote by phi of x.
When you use the notation phi maps D to the set E, this indicates that the values of phi always lie in the given set E, no matter what the x is in the domain. The smallest possible choice for such a set E, evidently, is the set of all values that phi takes as x ranges over the domain of D. The set of all these values is called the range of the function phi. And we say that the function phi is surjective, or onto, if for each y in the set E there is at least one point x in the domain such that phi of x is equal to y.
Now of course, from these definitions, you can logically prove the following fact, which I invite you to check, that, if you say that a function phi maps D to E, then it is surjective precisely when the range of phi is equal to E. Here’s an example. Consider the function which takes any real number x and gives you as its value of the function the value x squared. This is a function from R to the closed interval 0, infinity, and it’s surjective when it’s viewed that way because, for every y greater than 0, there is an x such that f of x– that is, x squared– is equal to y.
In fact, you can see there will generally be two such values of x, the square root of y and minus the square root of y. Now, we say that a function phi is injective, or synonym one to one, if for each y in its range there is exactly one point x such that phi of x is equal to y. So not more than one. As you can see, the example above is a function f that is not injective because, generally speaking, there will be two values giving y. For example, both minus 2 and plus 2 when you square them give you the value 4. However, such a thing can be fixed, and this is an important little step.
We can modify the function f– perhaps we even give it a new name, let’s call it g– by restricting its domain, as we say. That is, we consider that the domain now is just the set 0, infinity and not the whole real line as it was before. With this smaller domain, the new function g, whose formula is still x squared, now becomes injective. Here’s the picture that explains what we’ve done. We had this function, and we noticed that for positive values of y– y equal y hat– there were two values of x that led to the function having the value y hat– namely, square root of y hat and minus that.
But now, if we consider that the domain is restricted to only positive values of x, now there’s only one value of x for which you get y hat. The function now becomes one to one, or injective. Now, when the function phi has both properties that we’ve just described– it’s injective and it’s surjective– then we say it is bijective. And that’s the property we need in order to be able to define what is called the inverse function. The function phi to the power minus 1– normally read phi inverse– is the function that maps in the opposite direction. It maps the set E back to the set to D.
And the way it works is that phi inverse of a point y in E is the unique point x in the domain D having the property that phi of x equals y. See there is such a point by surjectivty, and there’s a unique such point by injectivity. And that’s why this definition now makes sense.
To say it again, to say that phi inverse of y is x is equivalent to saying phi of x equals y. The effect of phi and phi inverse is to cancel the other function out. That is, if you apply phi to a point x in the domain and then you apply phi inverse, you come back to x. And the other way as well. If you take a point y in the set E and you apply phi inverse of y, and then you take phi of that, then you come back to y. So the two functions cancel one another out and their effect. They’re inverse functions. Let’s go back to our example.
We had the function g mapping the closed interval 0, infinity to itself, and its rule is x squared. And we had observed that this function is both injective and surjective, so it admits an inverse function. This will be a function that maps 0, infinity to itself. And g inverse of y will be the unique x such that g of x equals y. Well, that will be the positive square root of y. And this function, then, is the inverse function to the original g. And as we see, one effect is to cancel out the other effect. For example, the square root of x squared will be x when x is positive.
And the square root of y all squared will be y. Now, graphically, you would expect there to be some connection between a function and its inverse, and it’s essential that we see and understand that connection. Let’s do it first for our example. Here’s the inverse function, the square root of t. There’s its graph. There is the original function, x squared but restricted to positive x’s. We tend to discern a rather great similarity between these two curves, do we not? What exactly is this similarity? Well, let’s explore it. If you go out a horizontal length t along the t-axis in the graph of root t, what point on that graph corresponds to that value of t?
Answer, it’s a certain point whose vertical height is the square root of t. Now, suppose you take this length t and you place it vertically along the graph of the function x squared. What point does that determine on that graph? Answer, it corresponds to the value of x for which you get t– that is, x is the square root of t. And as you can see then, this horizontal distance on the graph of x squared corresponds to the vertical distance on the graph of root t. In other words, looking at these x and y components, vertical has become horizontal and horizontal has become vertical. Otherwise, the things are the same. There’s just a transformation of a geometric type.
Now, if we want to go back and make horizontal vertical and vise versa, we could take our original graph of x squared. We could rotate it to switch vertical and horizontal. But now, there’s a problem. Namely, we like our vertical axis to be pointing upwards and not downwards. So we do a flip, a reflection about the x-axis, and we come back to what we recognize as the graph of the function of square root of t, the other graph. So here are the two functions, the function x squared and its inverse– or the function square root of x and its inverse– graphed simultaneously together on this picture.
There’s another way to recognise how to get one graph from the other which is important to know, and it refers to the following fact– that, by definition of inverse, a point x, y lies on the graph of the original function if and only if the point y, x lies on the graph of the inverse function. Now, this relationship, x, y to y, x, is something that is called reflection– reflection about the line y equals x. If you do this operation on one point, you get the other point on the other side of that line. And if you do it once more to this new point, you come back to where you started.
So to summarize, when you have the graph of an inverse function, it’s simply the reflection of the original graph about the line y equals x. We’ll come back to this in later discussions.