Testing if a function has an inverse.

Question

I was just wondering how you apply the rule: $$f(x_1) = f(x_2) => x_1 = x_2 $$ on the function: $$f(x) = x^3 - 9x^2 +33x +45$$

Any suggestions on how to proceed would be appreciated.

EDIT: Yes there should have been a $$9x^2$$ in there.

Answer 1

The derivative of $f$ is $$f'(x)=3x^2-18x+33=3(x^2-6x+11)=3[(x-3)^2+2]$$ which is positive. Therefore, $f$ is strictly increasing and hence injective. Nevertheless, an algebraic expression for the inverse it is complicated, because it involves solving of a cubic equation.

Answer 2

Very simple assume that $f(x_1)=f(x_2) \implies x_1^3+24x_1+45 = x_2^3+24x_2+45$ then we can simplyfy this by saying that $x_1^3-x_2^3 +24(x_1-x_2) =0$

If $x_1 \neq x_2$ then can not be true,thus they must be equal,thus function is injective thus has inverse