# Inverse of Functions

May 9th, 2015
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This is compact study of how to deal with questions related to Inverses of Functions.

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INVERSE FUNCTIONSConditions for being a function and having inversesFor being a function an equation has to pass the vertical test, i.e. every value of x in that expression should have a unique y.For example:So, whatever value we put in x, there is a unique f(x)/y. If x=2, we will get f(2)= 5 and thus for x=2, we wont be getting another value.Therefore, f(x) is a function.But when,So, whatever value we put in x, there are 2 possible values of f(x)/y. If x=4, we will get. So, we can say that, there is no unique f(x)/y. Therefore, is not a function.Now, in order for the function to have an inverse( being a one-to-one function), the function should pass the horizontal test, i.e. every value of x should have an unique y, which no other value of x cannot have.For example:When x=3, we get y=10 and we wont be getting y=10 for any other values of x. So, f(x) has an inverse or is a one-to-one function.But when, When x=3, we get y=9 and also when x=-3, we get y=9. So, 2 values of x have the same y. Thus, f(x) does not have an inverse or is a one-to-one function.Another way of finding if a function is one-to-one or not is by the following:Leads to For example,

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