Question

True or false: for any function, x=f^-1(y), then y=f(x)

Answer 1

YES

We can work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:

The function: f(x)=2x+3
Put "y" for "f(x)": y=2x+3
Subtract 3 from both sides:
y-3=2x
Divide both sides by 2:
(y-3)/2=x
Swap sides: x=(y-3)/2
Solution (put "f-1(y)" for "x") : f-1(y)=(y-3)/2

Answer 2

The given statement is false.

Explanation:

Given : For any function,
`x=f^(-1)(y)` then
`y=f(x)`

To find : The above statement is true or false?

Solution :

In the above statement the condition
`x=f^(-1)(y)` then
`y=f(x)` is valid for some function not for all.

Which means the statement is not true.

Taking a contrary example,

A trigonometric function

The function
`y=\sin x` is one-one and onto in the domain
`[-(\pi)/(2),(\pi)/(2)]`

Thus, its inverse exists in
`[-(\pi)/(2),(\pi)/(2)]`

i.e.,
`\text{In }[-(\pi)/(2),(\pi)/(2)],\ y=\sin x \Rightarrow\ x=\sin^(-1)(y).`

It depends on the domain for the given statement to be true.

Therefore, The given statement is false.