Not every function has an inverse. Whether or not a function has an inverse depends on whether it is one to one or many to one. A one to one function is any function which returns a single value of
for each value of
and a single value of
for each value of
Functions that are one-to-one satisfy both the ‘vertical line test’ - no vertical line can meet the graph more than once - and the ‘horizontal line test’ - no horizontal line can meet the graph more than once.
Every function must pass the vertical line test, else it is not a function. Those functions which do not pass the horizontal line test also are not one to one and do not have inverses.
We can often restrict the domain to a set of x values on which the function does pass the horizontal line test. On this domain the function will have an inverse.
On the diagram above,
passes the horizontal line test, but
and
do not.
We can restrict the domain ofto
– other intervals are possible – and the domain of to the interval
and with these restrictions bothandhave inverses.