Inverse for one-to-one functions This area takes functions, but where we usually take values of x and look at the corresponding values of f(x), here we take values of f(x) and look at what value of x produces this. Inverse functions with one to one mappingLet's look at an example: If we have the function, f: x → 3x, and the 'domain' x ∈ {1, 2, 3} Then we see the mappings are for this function over these x values is: So now we have the mapping of x to f(x). But we can also go backwards and map the range values (i. e. the values produced by f(x)) to those in the domain (i. e. the values of x used). So forward mapping gives 1 → 3 2 → 6 3 → 9 And inverse mapping gives 3 → 1 6 → 2 9 → 3 Looking at the inverse mapping, the values produced can also be written as another function: x → x/3, where x → {3, 6, 9}. This reverse mapping is a one-to-one mapping and is called the inverse function of f where f: x → 3x. The symbol for any inverse is f−1. So, f−1 x → x/3, x ∈ {3, 6, 9} is the inverse of f x → 3x, x ∈ {1, 2, 3} The relationship between the graphs of f and the inverse f−1 is shown in the diagram: From the diagrams you can see that the transformation to get from f(x) to f−1(x) is a reflection in the line y = x. This helps us to find the inverse of more complicated functions, and we do so by: Writing the equation as y = f(x). Swapping the letters x and y. (This is the same as reflecting in the line y = x.)Rearranging the formula into a new y = f(x). This is the inverse function. Example: Find the inverse function of Therefore: Rearrange to get, xy + 2x = y2x = y(1 − x)This means that the inverse function is, Inverse functions with one-to-many mappingThe above example had a 'one to one' mapping (see lesson 1 - mapping). If you have a one to many mapping this causes complications. This is because a single value of f(x) can be generated from many different values of x and this cannot be defined using a single inverse function. The way we can get around this is to set the domain (the range of x values the function can use), such that only one value of x will produce one value of f(x). This is quite a complex idea, so let's look at an example. Using the function x2 the rearrangement gives us, f−1(x) = ±√ xThis would then define a one-to-many mapping and therefore not give a function (as a function cannot be a one-to-many mapping). Therefore, f: x → x2, x ∈ R does not have an inverse function. You can obtain the reverse mapping by only allowing x to take positive real numbers (or only negative real numbers). So we have, f: x → x2, x ∈ R+ which is a one-to-one mapping. The reverse mapping only allows positive square roots in the range. So, the inverse is, f−1 : x → √ x, where x ∈ R+ Graphs of Inverse FunctionsAs mentioned earlier, all you need to do to sketch the graph of the inverse function f−1 is to reflect f in the line y = x. However, if f does not have an inverse, you will still be able to reflect the graph, but it will not represent the inverse function. Example: Find the inverse function of f(x) ≡ 3x, x ∈ R, and sketch the graph. To find f−1 we have to map values of 3x back onto the values of x. Therefore: for f(x), y = 3x For f−1(x), x = 3y, and by taking logarithms we get, f−1(x) ≡ log3x The curves of the two function are the same, but the inverse function has been reflected in the line y = x. See the diagram below: