Posted by anonymous on Wednesday, January 9, 2013 at 6:01pm.
Surely you mean
f(x) = (2x+8)/(x+6) and g(x) = (6x-8)/(2-x)
f(g(x)) = (2(6x-8)/(2-x) + 8)/((6x-8)/(2-x) + 6)
multiply by (2-x)/(2-x)
= (2(6x+8) + 8(2-x))/(6x-8 + 6(2-x))
= (12x-16 + 16 - 8x)/(6x-8 + 12 - 6x)
= 4x/4
= x
check:
let x=4
g(x) = 16/-2 = -8
then f(g(4))
= f(-8) = (-16+8)/(-8+6) = -8/-2 = 4
I will leave it up to you to find g(f(x))
but you should also get x as a final answer
If f(x) and g(x) are any functions so that
f(g(x)) = x, then the two functions are inverses of each other.
We have that situations, so they are inverses of each other
or .... the long way ... actually find the inverse of one of them
let y = (2x+8)/(x+6)
inverse is
x = (2y + 8)/(y+6)
xy + 6x = 2y + 8
xy - 2y = 8 - 6x
y(x-2) = (8-6x)
y = (8-6x)/(x-2) = (6x-8)/(2-x) which is the other function.
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