## inverse f(inverse g(x)) = inverse g(inverse f(x))

Quadratic equations and inequalities, variation equations, function notation, systems of equations, etc.
daz1990
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Joined: Tue Oct 20, 2015 2:24 pm
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### inverse f(inverse g(x)) = inverse g(inverse f(x))

Hi,

I am stuck about two thirds of the way through this question and can't figure the remaining part out.

I have;

f(x) = 24x-15 g(x)=2x-d where d is an unknown value, it is known that inverse f(inverse g(x)) = inverse g(inverse f(x))

I have to first of all find the value of d, and secondly find; inverse f(inverse g(20))

We have been using mathCAD to solve similar but I really want to learn how to solve this the "old school" way using a pen and paper!

So far I have got;

inverse f(x) = x/24 - 5/8 inverse g(x) = x/2 + d/2

next step...

inverse f(inverse g(x)) = (d/2 +x/2)/24 - 5/8 = d/48 + x/48 - 5/8

inverse g(inverse f(x)) = d/2 + (x/24 - 5/8)/2 = d/2 + x/48 - 5/16

Therefore; d/48 + x/48 - 5/8 = d/2 + x/48 - 5/16

I am now stuck at that point and cannot figure out the next steps. On mathCAD the next step is to (solve, d) and it gives you the answer, and then for the second part of the question you can substitute the values of d and x into the equation(s) to give you the answer.

I have been stuck on this for hours and just cannot get past the final hurdle, please help!! P.s. I do know the answers from mathCAD which are; d = -15/23 and for the second part of the question the answer is -0.222.

maggiemagnet
Posts: 358
Joined: Mon Dec 08, 2008 12:32 am
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### Re: inverse f(inverse g(x)) = inverse g(inverse f(x))

f(x) = 24x-15 and g(x)=2x-d where d is an unknown value, it is known that inverse f(inverse g(x)) = inverse g(inverse f(x))

I have to first of all find the value of d, and secondly find; inverse f(inverse g(20))

We have been using mathCAD to solve similar but I really want to learn how to solve this the "old school" way using a pen and paper!
I don't know anything about "mathCAD" but let's see about the old-school way!

They show how to find inverse functions here. I get the same inverses as you. They gave you that the compositions of the functions' inverses are equal. I get the same results as you:

d/48 + x/48 - 5/8 = x/48 - 5/16 + d/2

You can subtract x/48 from both sides to get:

d/48 - 5/8 = d/2 - 5/16

This is just a one-variable linear equation (here). Multiply through to clear the denominators:

d - 30 = 24d - 15
-15 = 23d

Where are you getting stuck in finishing this? Thanks!

daz1990
Posts: 5
Joined: Tue Oct 20, 2015 2:24 pm
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### Re: inverse f(inverse g(x)) = inverse g(inverse f(x))

Thank you so much!

I think I just had a mind block on this particular problem after a whole day of studying, the more I was looking at it the worse it was getting, but in fact it is simple now reading through your notes!

I wasn't multiplying the whole problem by the LCD of 48 for some reason, so simple

Thanks again