Determine The Inverse G(X) Of The Function F(X) = (1+(4/X)), Stating Its Domain And Range. Verify That F(G(X)) = G(F(X))=X And That G′(F(X)) = (1/(F′(X))

[chin123456]

Calculus > inverse function

[Peter Pedant]

oh come on come on
we wont do your prep for you
see FF 's link
half way down .....
gx = y = 4 / (x-1)
now solve for x - ie get x in terms of Y
I think I can see a - - - - cross multiplication

hi boys - having a gd evg? - shocking news about the PM innit?

[douglas9401]

And with just a few words I'm back in maths hell.

Those who can do, those who can't sweat, until they realise that not everybody can do everything. :-)

[Mamyalynne]

Peter - this was the 'Car polishing' thread.

https://www.theanswerbank.co.uk/Technology/Question1700465.html

[Peter Pedant]

oh thanks mama

[fiction-factory]

Hi PP.
Yes. Start by writing f(x) as y=1+(4/x)
So y-1=4/x
so x = 4/(y-1)
so to find the inverse g(x) change x to y and y to x. *
So g(x) is 'y' =4/(x-1)

You can check it by putting in some numbers. eg try x= 4
f(x)= 1+(4/4)=2. Think of that as the output.
Use that output as the input into the inverse, g(x) and we get g(x)=4/(2-1) =4, so gets us back to where we started

Now try x=0.5
f(x)= 1+(4/0.5)= 9
g(x)=4/(9-1)= 0.5.
Ta da!

Works for any initial input value (except x=0 since can't divide 4 by 0 )

*I know jim probably wouldn't approve of this GCSE type method for 'doing' these whereby you swap x and y round but it gets to the answer.

Anyway- must leave it there as I've been furloughed and am not allowed to do this stuff.

[jim360]

// *I know jim probably wouldn't approve of this GCSE type method for 'doing' these whereby you swap x and y round but it gets to the answer.//

I do it exactly the same way :P

[fiction-factory]

Okay, if anyone else is trying to refresh their memory on these:

Now we know g(x) and f(x), we can do step 3 which is to find g(f(x)) and f (g(x)). We can do this using algebraic substitution
To find g (f(x)) you apply the g(x) function to f(x).
So take g(x), which is 4/(x-1) and replace the x by f(x) which we were told is 1+(4/x) and we get our answer of....x.
Rinse and repeat for f(g(x)