Functions are a fundamental part of mathematics. They provide connections from numbers to numbers. We’re going to discuss them in some detail. We’re going to talk about functions that send the real numbers to the real numbers. What’s a function? It’s nothing but a rule. We could call a function f. f is a rule that takes a given number that we call the argument and assigns to it another number that we call the value of the function. Consider this example. f is the function which, when given a number, doubles it and adds 1. So for example, f will assign the value 7 to the argument 3 because doubling 3 gives you 6 and adding 1 then gives you 7.
We can express the rule behind the function symbolically as follows– f of x equals 2x plus 1. Now, it’s important to realise that in such an expression the x is just a placeholder. That’s what we call a dummy variable. There is no x, there is no spoon. I could equally have called the argument u in this defining expression. I could have said that f of u equals 2u plus 1 or f of whatever equals twice whatever plus 1. But it’s handy to have such an expression because you can then replace the dummy variable by a specific value, like 9, and work out the value that corresponds to 9, namely 19. More notation.
When we write this kind of expression, we often read it as saying, the function f maps x to 2x plus 1. The word “maps” here is just another way of saying “sends to.” So here’s our function. And now here’s another function that I call g. g will take a number and square it, so g of 3 is 9, for example. If we have some functions, we can create new ones with certain arithmetic operations. For example, we can take the sum or the product of two functions. The sum of f and g is the function that assigns to x the sum of the two values– f x and g x.
In our specific example here, we would get this answer for the defining formula. Similarly, the product multiplies the 2 function values. When you work that out, you get the defining expression for fg. The composition of two functions means that you’re going to apply the functions in turn. For example, f composed with g means that first you apply g to x and then you apply f to the answer g of x. In our example, this would give us the following result– 2x squared plus 1. We can also compose the two functions in the opposite order.
The function g composed with f would give us f of x squared, which when worked out gives us this expression, which we notice is different from the composition of f with g above. In other words, the order makes a difference. The composition operation is not a commutative one. You’ll get a different answer, depending on the order. So we have f, we have g. We can also form the quotient f over g. In our specific example, this would give us the defining rule that you see.
And now a new consideration comes to the fore because you see that this formula presents a problem when x is 0, because then the denominator would be 0 and we don’t want to divide by 0, it’s undefined. So in other words, the quotient is naturally restricted to those numbers x that are different from 0. That is, the set R minus or delete 0. More generally, any given function phi has a subset associated with it called its domain– often called capital D– the set in which we define the function. We used a notation that you see to express that the function phi maps D to the reals. The domain of a function can sometimes be prescribed, it can be given.
For example, we might wish to consider our function f of before only for positive values of x. That’s perfectly all right. Often though the domain is implicit. It’s taken to be the largest set upon which the defining formula makes sense. Here’s an example. Observe the defining formula for this function. What do you think the natural domain of the function is? Well, you see that the formula presents a problem when x equals 1 or x equals 2. If you delete those two points from the reals, you’ll have the natural domain of the function. The notation that you see is read as phi maps D to the set E. I’ve introduced a new set E here.
This notation indicates that the values of phi always lie in the set E. What is the smallest possible choice we could make for a set E of this type? Well, the smallest possible choice, called the range, would be the one that includes all the possible values of phi as the argument x varies over the domain D. And that’s the definition of range. Here’s an example. I define a function h, as you see. Now, we do remember that the defining formula is not the function. It just gives a way of remembering how the function operates. The function is a rule. We want to find the domain and the range of this function h.
As regards the domain, we see that it suffices to avoid the value x equals 1. So we find the domain to be R delete 1. What about the range though? Can you see that the values of h are always strictly positive numbers from the very nature of the formula? So therefore, it’s clear that the range of h will be included in the strictly positive numbers. I claim that every strictly positive number will be in the range. Let’s show that now. Let r be any strictly positive number. Then 1 over r is also a strictly positive number.
And there certainly is an x for which absolute value of x minus 1 equals that 1/r, because that means the distance from x to 1 is 1/r. If you take reciprocals here, you get an expression which you can recognise as saying, h of x equals r. In other words, every strictly positive r is the image under the function h of some x. So the range therefore is the set of all strictly positive numbers.