Question

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

Answer 1

Answer:

The given statement is false.                      

Step-by-step 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$[-\frac{\pi}{2},\frac{\pi}{2}]$

Thus, its inverse exists in $[-\frac{\pi}{2},\frac{\pi}{2}]$

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

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

Therefore, The given statement is false.                                                                    

Answer 2

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