1 It is important to note that not all functions will have an inverse function Since the inverse f z takes an output of f and returns an input of f in order for f to itself be a function then each output of f input to f must correspond to exactly one input of f output of f in order for f to be a function You might recall that this is the definition of a one to one function Properties of inverses In order for a function to have an inverse it must be a one to one function That is its graph must pass the Horizontal Line Test In some cases it is desirable to have an inverse for a function even though the function is not one to one In those cases we can often limit the domain of the original function to an interval on which the function is one to one then find an inverse only on that interval If you have not already done so go back to the toolkit functions that were not one to one and limit or restrict the domain of the original function so that it is one to one If you are not sure how to do this proceed to Example 6 Example 6 The quadratic function h x x is not one to one Find a domain on which this function is one to one and find the inverse on that domain Show solution 4 Exercise 1 6 5 The reciprocal squared function f x r function is one to one and find the inverse on that domain using the variable Domain 0 00 0 is not one to one Find a positive domain on which this uring an incorrect one