Math Help

[shadowcalkins]

My question is: The Martians ask you to explain one last thing, Ultimate Math Ambassador. Assign any number to x. Using complete sentences, explain whether f(g(x)) and g(f(x)) will always result in the same number. You will use the inverse function that you created in problem number 5 for g(x).

The inverse function I have is :
x-6 divided by 2 = y.

The original equation was: y=4x+4-2x+2

I cannot for the life of me figure this out please help!

[jim360]

Explaining in complete sentences is a bit tough, but essentially the idea is:

"Yes, because in applying a function and then its inverse, you are doing something to that number and then exactly undoing it again, so that you get the number you started with. In the meantime, The "inverse of the inverse" is the original function itself, so you can apply the same logic to applying the inverse function followed by applying the original function. This is equivalent to undoing something to a number and then doing it again, processes which again cancel out. Hence for f(x) and g(x) as defined, we can expect f(g(x)) to be equal to g(f(x)) for all x, and in fact f(g(x)) = x."

[mac44]

I found this,

Let g(x)=3x+1 for instance. f(g(x))=2(3x+1)+3=6x+5. g(f(x))=3(2x+3)+1=6x+10. No, they do not have to result in the same number. If we remove the m value from y=kx+m with arbitrary k values, however, they will. The same goes for f(x)=x^a and g(x)=x^b.

[jim360]

The only problem with that answer, mac, is that it doesn't use the functions he is told to use. In general it's true that f(g(x)) is not equal to g(f(x)), but then here g is specifically defined as f-inverse, so they must be equal.

I've noticed that my answer might be just a bit too general for the problem, so a better answer might be: "Yes, they must be the same. In the first case, f(g(x)), we subtract 6 from x, then halve the result, and then double that and add six to get back to where we started. In the second case, g(f(x)), we first double x, then add six, then subtract six again, and half that, so once again everything cancels and the two are the same."

Hope this helps.

* * * * * * *

Don't use this answer, but you could also say something like: "Hang on! You're a Martian Ambassador who has travelled all the way to Earth! I doubt you managed that without knowing how inverse functions work..."

[Buenchico]

Lets start by simplifying the original equation to y = 2x + 6.
i.e. the function f doubles whatever it's applied to and adds 6.

You correctly have the function g defined as subtracting 6 from what it's applied to and then dividing the answer by 2.

Hence g(f(x)) = ((2x + 6) - 6)/2 = 2x/2 = x

and f(g(x)) = 2((x - 6)/2) + 6 = (2x - 12)/2 + 6 = x - 6 + 6 = x

Thus g(f(x)) = f(g(x)) for all values of x