Question

Consider the functions

f(x)=2x+8/x+6; g(x)=6x-8/2-x
Find f(g(x))
Find g(f(x))
Deterrmine if f and g are inverses of each other.

a. What is f(g(x))?

Give any values of x that need to be excluded from f(g(x))

x=?

b. what is g(f(x))?

ive any values of x that neeedd to be excluded from g(f(x))

x=?

c. Are the functions f & g inverses of each other ?

**Please show work**

Answer 1

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.