## Inverting a Function

Given a function we can easily find a value of given a value of by substituting the value of into the function If however we want to find the value of given a value of then at some stage we will have to either invert the function or solve an equation to find The first method is usually preferable because it is general: in inverted function means we can find any value of given a value of There are three steps to inverting a function 1. Make the subject so that you have another function 1. Interchange occurrencesof and sono you have 2. Replace by :the answer is Example: Find 1. 2. Interchange and : 3. If you draw the graphs of and on the same axis you will notice something very striking. The line is a line of symmetry: to obtain the graph just reflect the graph in the – axis. To see why this is so, notice that steps 1 and 2 above interchange and  This is illustrated above for the graphs and which are inverse to each other.

A problem may arise if you have a function which gives the same value of for more than one value of When you try to invert the function and you find a value of may return no value of or more than one value of It is necessary in a case like this to restrict the domain of the inverse function to eliminate those “impossible” 's and “duplicate” 's. For example, if – we take the positive square root to ensure only one value of for each value of and we must have If then and the domain of is We take the range to be so that there is one value of for each  