## Inverting a Function

Given a functionwe can easily find a value ofgiven a value ofby substituting the value ofinto the functionIf however we want to find the value ofgiven a value ofthen at some stage we will have to either invert the function or solve an equation to findThe first method is usually preferable because it is general: in inverted function means we can find any value ofgiven a value of

There are three steps to inverting a function

1. Makethe subject so that you have another function

Interchange occurrencesofandsono you have

Replaceby:the answer is

Example:Find

Interchangeand:

If you draw the graphs ofandon the same axis you will notice something very striking. The lineis a line of symmetry: to obtain the graphjust reflect the graphin the– axis. To see why this is so, notice that steps 1 and 2 above interchangeand

This is illustrated above for the graphsandwhich are inverse to each other.

A problem may arise if you have a functionwhich gives the same value offor more than one value ofWhen you try to invert the function and you finda value of may return no value ofor more than one value ofIt 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 offor each value ofand we must haveIfthen and the domain ofisWe take the range to beso that there is one value offor each