Answer to Question #233433 in Discrete Mathematics for kannan

Question #233433

whether each of these functions is a bijection from R to R. f (x) = x 3 -1 


1
Expert's answer
2021-09-06T14:54:39-0400

Consider the function f:RR, f(x)=x31.f:\R\to\R,\ f(x)=x^3-1. Let us show that this function is an injection. Let f(x)=f(y).f(x)=f(y). Then x31=y31,x^3-1=y^3-1, and hence x3=y3.x^3=y^3. It follows that x=y,x=y, and ff is an injective function. Further, let us show that this function is a surjection. Since for any yRy\in\R the equation f(x)=yf(x)=y is equivalent to x31=y,x^3-1=y, and hence has the solution x=y+13R,x=\sqrt[3]{y+1}\in\R, we conclude that ff is a surjection. Consequently, the function ff is a bijection.


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