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Chapter 5: Q. 54 (page 269)

The function fis one-to-one. Find its inverse and check your answer. Graph f, f-1, and y = x on the same coordinate axes.

f(x)=x3+1

Short Answer

Expert verified

The inverse of the function is f-1(x)=x-13.

Step by step solution

01

Step 1. Given information :

The given function is one-to-one function.

f(x)=x3+1.

02

Step 2. Finding the inverse of the function :

For finding the inverse of the function we need to interchange the variables.

let, f(x)=y=x3+1

Interchanging the variables x, y,

x=y3+1y3=x-1y=x-13

So, the inversef-1(x)=x-13.

03

Step 3. Check the solution :

To check whether the function is inverses are not,

let, f(f-1x)=x

f(x-13)=xwegot,f-1(x)=x-13(x-13)3+1=xgiven,f(x)=x3+1(x-1)+1=xx=x

Similarly, f-1(fx)=x

localid="1647509029875" f-1(x3+1)=xgiven,f(x)=x3+1(x3+1)-13=x,f-1(x)=x-13x33=xx=x,f-1(x)=(-鈭�,鈭�)henceproved.

04

Step 4. Graph representation of f,f-1 and y=x :

Graph ;

Domain of f= range of localid="1647508985116" f-1=(-鈭�,鈭�)

Domain of f-1= range of localid="1647508995977" f=(-鈭�,鈭�).

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