The inverse of a function
Define inverse function. Give a criterion for a function to have an inverse. Solve the following problems.
(1) Prove that the function f:R-->R, f(x)=3x+4 has an inverse, and find it.
(2) The function f:R-->R, f(x)=kx^2+3x+4 has an inverse. Find k.
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Inverse of a function
Let f:A-->B and g:B-->A be functions, where A and B are some sets of real numbers. f is called an inverse of g (and g is called an inverse of f) if the compositions fg and gf are the identity functions, i.e. , for any number x from A, we have g(f(x))=x , and ,for any number y from B, we have f(g(y))=y .
Not any function has an inverse. For example, consider the function f:R-->R, f(x)=x^2 , where R is the set of real numbers, and assume that it has an inverse g. Then f(g(-3))= -3. But f(g(-3))=〖(g(-3))〗^2, and hence ...
Solution Summary
The solution of an inverse function is given, as well as a criterion for a function to have an inverse. The following problems are solved with detailed explanations.