Kenneth Rosen Edition 7 Exercise 2.3 Question 69 (Page No. 155)

Question

Find the inverse function of $f(x) = x^3 +1.$

Answer 1

Given $f(x)= x^3 + 1$

Let $y=f(x)$

$y=x^3+1$

$x=y^3+1$

$y^3=x-1$

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

Answer 2

The inverse of function:

• $f$ is a bijection ,$f_{A\rightarrow B}$
• $f^{-1}$ is a bijection ,$f_{B\rightarrow A}$
• $f^{-1}_{B\rightarrow A}(x) = y$   iff  $f_{A\rightarrow B}(y) = x$

Given that $f(x) = x^{3} + 1$

$f^{-1}(x) = y \ $ iff $\ f(y) = x$

$\implies y^{3} + 1 = x$

$\implies y^{3} = x - 1$

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