88.8k views
0 votes
The function f(x)=x^3+2 is one to one Find part B

The function f(x)=x^3+2 is one to one Find part B-example-1

1 Answer

2 votes

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^{(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)^{(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^{(3)/(3)}\text{ = x}

This is the final part.

The function f(x)=x^3+2 is one to one Find part B-example-1
User Innom
by
8.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories