math help (f o g)^-1 (x)?

Question

(f o g) ^ -1 (x)
How do you solve that?
I know (f o g) (x) = f ( g (x) )
But, I don't know about (f o g) ^ -1 (x) HELP

Answer 1

Hi, Gerard.



It might be helpful for us to look at this in concrete terms. For example,



Suppose f(x) = x + 3, and

g(x) = 2x



Then, f(g(x)) = 2x + 3 = (f ○ g)(x)



Now, if we want to find this function's inverse (f ○ g)^-1(x), suppose y = (f ○ g)(x), and then solve for x:



y = 2x + 3

=>

2x = y - 3

=>

x = (y - 3) / 2



Swapping variables x and y,



y = (x - 3) / 2



So, (f ○ g)^-1(x) = (x - 3) / 2



But, consider that f^-1(x) = x - 3, and

g^-1(x) = x/2



So, (x - 3) / 2 is actually g^-1(f^-1(x)) = (g^-1 ○ f^-1)(x)



And, (f ○ g)^-1(x) = (g^-1 ○ f^-1)(x)



Still, you have to be careful. Consider the function h(x) = x^2. Then, we'd suspect its inverse is ±√x. However, since this doesn't pass the horizontal-line test, it isn't a function, unless we limit its domain. So, we'd have to say h^-1(x) = √x, defined over x ≥ 0.

Answer 2

Fog 1

Answer 3

if both f and g have inverses then:



(f o g)^-1 = g^-1 o f^-1.



to see this, let's compose g^-1 o f^-1 with f o g:



(g^-1 o f^-1)o(f o g)(x) =



(g^-1 o f^-1)(f(g(x)) =



g^-1(f^-1(f(g(x)))) = g^-1(g(x)) = x



(to "undo" what f does to what g does to x,



undo what what f did to f(g(x))



(this is f^-1(f(g(x))), ok?), and then undo what g did to g(x),



(this is g^-1(g(x)) = x).).



perhaps a picture will make more sense:



x--->g(x) = y---> z = f(y) = f(g(x))



that is what f o g does.



so to undo it, we go backwards:



z-->y-->x



what takes z-->y? that is f^-1



what takes y-->x that is g^-1



do f^-1 first, and then g^-1, is g^-1(f^-1(z)) = g^-1 o f^-1(z)



since this undoes what f o g does, it must be (f o g)^-1.