The domain in which the given function becomes one-to-one and find the inverse of the function with the restricted domain.

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2.7, Problem 75E
To determine

To find: The domain in which the given function becomes one-to-one and find the inverse of the function with the restricted domain.

Expert Solution

The domain in which the given function is one-to-one is [0,2] and the inverse of the function is f1(x)=4x and domain is [0,4] .

Explanation of Solution

Given information:

f(x)=4x2

Concept used: A function with domain A is called a one-to-one function if no two elements of A have the same image, i.e.

f(x1)f(x2)    Whenever x1x2

If f be a one-to-one function with domain A and range B. Then its inverse function f1 has domain B and range A and is defined by

f1(y)=x      f(x)=y

Calculation:

From the given graph of the function, it is clear that the value of f(x) don’t repeat in one side of the Y-axis. So, in the domain [0,2] the given function is one-to-one.

f(x)=4x2      0x2y=4x2             0y4x2=4yx=4y             [ 0x2, ignore (-ve) value]f1(y)=4y         0y4

Therefore, the inverse of the given function with restricted domain can be written as

f1(x)=4x           [0x4]

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