prove that f(x)=2x^2-3 and g(x)= √((y+3)/2) are inverses of each other

Question

prove that f(x)=2x^2-3 and g(x)= √((y+3)/2)  are inverses of each other

zzr0ck3r · Answer

This is not true in general. If you restrict the domain to $x\ge0$ then it is and you need to show that $$f(g(x))=x=g(f(x)) \text{ for all }x\ge0.$$

Hint: $f(g(x))=2(g(x))^2-3 = 2(\sqrt{\frac{x+3}{2}})^2-3=...=x$

Complete the missing steps and then do something similar for $g(f(x))$.

p0sitr0n · Answer

or you could switch $$f(x)$$ and $$x$$ places, then isolate the x and showing that the new function is equal to $$g(x)$$.

supersmart1001 · Answer

https://www.youtube.com/watch?v=crpeERDwYag

zzr0ck3r · Answer

definition of inverse, is that a*b=I=b*a where I is the identity and * is the operation. The operation here is composition which has identity function f(x) = x. 

So to prove it you need to show what I suggested. 

The difference here is as follows: When you switch x and y and solve you are not showing existence, you are only showing uniqueness. i.e. if the inverse exists, it is uniquely blah. To show existence you must show that the existence of an inverse function and to show that by definition. 

This is actually a great question about inverse. First we must not the restricted domain as above in order to call something an inverse because domain and range of an function and its inverse must be opposite, and you can see this problem when you try and do the switch thing. 

Finding the inverse of f(x) (the switch method) will lead to two answers which is what tells you that you must restrict the domain for this to work out.