Question

Let f : R → R be defined by f(x) = 3√(1 – x 3 ). a. Prove that f is bijective b. Determine f -1 (x)

Answer 1

a. Let $f(x_1)=f(x_2).$ It means that

$$\sqrt[3]{1-x_1^3}=\sqrt[3]{1-x_2^3}$$

$$1-x_1^3=1-x_2^3$$

$$(x_1-x_2)(x_1^2+x_1x_2+x_3^2)=0$$

$$x_1-x_2=0$$

$$x_1=x_2$$

The function $f(x)=\sqrt[3]{1-x^3}$ is bijective (one-to-one ) from $\R$ to $\R.$

$$f(x)=\sqrt[3]{1-x^3}, x\in \R$$

$$y=\sqrt[3]{1-x^3}$$

Change $x$ and $y$

$$x=\sqrt[3]{1-y^3}$$

Solve for $y$

$$y^3=1-x^3$$

$$y=\sqrt[3]{1-x^3}$$

Then

$$f^{-1}(x)=\sqrt[3]{1-x^3}$$