Question

The function f(x)=x^3+2 is one to one Find part B

Answer 1

a) The given function is expressed as

f(x) = x^3 + 2

The first step is to find the inverse of the function. We would replace f(x) with y. It becomes

y = x^3 + 2

The next step is to interchange x and y. We have

x = y^3 + 2

Next, we would solve for y. We have

y^3 = x - 2

taking the cube root of both sides of the equation,

$\begin{gathered} y\text{ = }\sqrt[3]{x\text{ - 2}} \\ \text{Changing y to f}^{-1}, \\ f^{-1}(x)\text{ = }\sqrt[3]{x\text{ - 2}} \\ \text{Note } \\ a^{\frac{b}{c}}\text{ = (}\sqrt[c]{a})^b \end{gathered}$

b) To show that f(f^-1(x)) = x, we would substitute x = the inverse function into the original function. We have

$f(f^{-1}(x))\text{ = (}\sqrt[3]{x\text{ - 2}})^3\text{ + 2 }=(x-2)^{\frac{3}{3}}+2=x-2+2=x^{}$

To find f^-1(f(x)), we would substitute x = the original function into the inverse function. We have

$f^{-1}(f(x))=\text{ }\sqrt[3]{x^3\text{ + 2 - 2}}=\text{ }\sqrt[3]{x^3}=x^{\frac{3}{3}}\text{ = x}$

This is the final part.