[Answer] Which function has an inverse that is also a function?

Answer: C. {(-1 3) (0 4) (1 14) (5 6) (7 2)}
Which function has an inverse that is also a function?

In mathematics an inverse function (or anti-function) is a function that “reverses” another function: if the function f applied to an input x gives a result of y then applying its inverse function g to y gives the result x i.e. g(y) = x if and only if f(x) = y. The inverse function of f is also denoted as ${\displaystyle f^{-1}}$. As an example consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely take a number x multiply it by 5 then subtract 7 from the result) to reverse this and get x back from some output value say y we would undo each step in revers…

In mathematics an inverse function (or anti-function) is a function that “reverses” another function: if the function f applied to an input x gives a result of y then applying its inverse function g to y gives the result x i.e. g(y) = x if and only if f(x) = y. The inverse function of f is also denoted as ${\displaystyle f^{-1}}$. As an example consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely take a number x multiply it by 5 then subtract 7 from the result) to reverse this and get x back from some output value say y we would undo each step in reverse order. In this case it means to add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by ${\displaystyle g(y)={\frac {y+7}{5}}.}$ With y = 5x − 7 we have that f(x) = y and g(y) = x. Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse it must have the property that for every y in Y there is exactly one x in X such that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f.

If it would be true the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant) then it has an inverse that is also a polynomial function. It is unknown whether this is true or false even in the case of two variables.

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