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We already know that a function is a set of pairs of numbers. If we reverse the numbers in the pairs we get a new function. Lets do so with the following function: f = (1, 2), (2, 4), (3, -1), (4, -2) We can see what happens and call this new set g: g = (2, 1), (4, 2), (-1, 3), (-2, 4) We have obtained a new function. However, this does not always work. Let f be the following set: f = (1, 2), (2, 4), (3, -1), (4, 2) Therefore g is: g = (2, 1), (4, 2), (-1, 3), (2, 4) which is not a function as g(2) is not determined by just one value; in other words g does not satisfy the condition of the function. There are two pairs of numbers, (2, 1) and (2, 4), whose first coordinate is the same and second coordinate is different. What is the difference between these two examples? The reason is that in the second example f(1)=f(4)=2 and when the pairs are reversed g(2) is not determined by just one value; therefore g is not a function. In the first example the different values of x give different values of y. Functions which behave in the same way as this first example are called injective or one-to-one. DEFINITION: A function f is injective or one-to-one if f(a) is distinct to f(b) when a is distinct to b. When another function is obtained by reversing the pairs of a function we can say that this function has an inverse. In the light of what we have stated above, the only functions which have inverse functions are injective functions. DEFINITION: If f is an injective function, the inverse of f ,denoted by f-1 is the set: f-1 = (a, b) (b, a) Î f In other words, f-1 = (x, y) x=f(y), if y is the domain of f = (f(y), y) if y is the domain of f From the definition we can see immediately that the domain of the inverse function f-1 is the range of f and, reciprocally, the range of f-1 is the domain of f. It is also easy to see that f-1(a)=b is the equivalent of f(b)=a. By using x and y which we often use when referring to functions f-1(x)=y is equivalent to f(y)=x. Another way of saying this is that: f(f-1(x))=x (where x belongs to the range of f), or, f-1(f(x))=x (where x belongs to the domain of f). By using the composition of functions, referring to the function defined by I(x)=x, I (Identity function) we can say that: fof-1 = I and f-1of = I except when the domain of the second member of these two equations is bigger than that of the first member if the domain of either f or f-1 is not all of R. Incidentally, if a function has an inverse, what will (f-1)-1 be equal to? i.e. the inverse of the inverse function? We have often referred to the idea of inverse functions in earlier units at this level. However, we have not labelled them as such. Think back to how we defined the square root, cube root etc. The following window represents the graphs of the two function f and g from activity 2. The functions are indicated in blue and the reverse pairs of numbers in pink. The control point moves along the functions indicating a pair of the function and the corresponding pair of the inverse function. The other straight line that you can see (in green) bisects the first and third quadrant (the straight line of the equation y=x).
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