Question

Find the inverse of the following function. Then prove they are inverses of one another.

f (x)= root 2x-1.

Answer 1

Answer:
`(x^2+1)/(2)`

Explanation:

Given

`f(x)=√(2x-1)`

We can write it as

`\Rightarrow y=√(2x-1)`

Express x in terms of y

`\Rightarrow y^2=2x-1\\\\\Rightarrow x=(y^2+1)/(2)`

Replace y be x to get the inverse

`\Rightarrow f^(-1)(x)=(x^2+1)/(2)`

To prove, it is inverse of f(x).
`f(f^(-1)(x))=x`

`\Rightarrow f(f^(-1)(x))=\sqrt{2* (x^2+1)/(2)-1}\\\\\Rightarrow f(f^(-1)(x))=√(x^2+1-1)\\\\\Rightarrow f(f^(-1)(x))=x`

So, they are inverse of each other.