The inverse, domain and range of inverse and verify that they are inverses of ea

OlmekinjP

OlmekinjP

Answered question

2021-09-14

The inverse, domain and range of inverse and verify that they are inverses of each other.
Given: The function f(x)=2x3

Answer & Explanation

dieseisB

dieseisB

Skilled2021-09-15Added 85 answers

Formula used:
To find inverse, interchange x and y and then make y as the subject of the formula.
To verify the functions are inverses of each other, find (fof1)(x) and (f1of)(x). If both are equal to x then the function are inverses of each other.
Calculations:
Here f(x)=2x3
Let y=2x3
Interchanging x and y we get
x=2y3
Dividing by 2 on both sides
x2=y3
Taking cube root
3{x2}=y
So f1(x)=3{x2}
We know that cube root function is defined for all real numbers. Therefore, the domain of f1=(,)
We know that cube root function can produce any real outcome. Therefore, the range f1=(,)
Now, (fof1)(x)=f(f1(x))=f(3{x2})=2(3{x2})3=2x2=x
(f1of)(x)=f1(f(x))=f1(2x3)=3{2x32}=x
Hence verified
Conclusion:
f1(x)=3{x2}
Domain: (,), Range: (,)

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