Question

Prove if the given functions are inverses by evaluating the composition f(h(x)) or h(f(x))

g(x) = -(4/-x-3) +1

f(x)= (3/-x-1) -2

How do we go about once we for example plug in the first expression into x in the second expression, in terms of what to do exactly.
Thanks

Answer 1

First, make sure you have typed things correctly. What you have written is gibberish. So, I shall go with
g(x) = -4/(-x-3) + 1
First of all, what's with all those - signs? Oh, well ...
swap variables and solve for x:
h(x) = 4/(x-1) - 3 = (7-3x)/(x-1)

By definition, f^-1(f(x)) = f(f^-1(x)) = x
So, to show that g and h are inverses, just evaluate g(h) and h(g)
g(h(x)) = -4/(-h(x)-3) + 1
= -4/(-(7-3x)/(x-1)-3)+1
= 4/((7-3x)/(x-1)+3)+1
= 4/((7-3x+3x-3))/(x-1))+1
= 4/(4/(x-1))+1
= 4 * (x-1)/4 + 1
= x-1+1
= x
Work h(g(x)) the same way

After you fix up your functions with the proper use of parentheses, of course!

Answer 2

oh. sorry, what exactly happened at the “swap variables and solve for x” step? How did you get (7-3x)/(x-1)

Answer 3

Ooops my bad. If you swap first, solve for y
you need to review how to find an inverse function.
If y = 2x-3
to find f^-1, first swap variables
x = 2y-3
then solve for y
y = (x+3)/2

Or, you could solve for x and then swap
y = 2x-3
y-3 = 2x
(y-3)/2 = x
x = y^-1 = (x-3)/2

Answer 4

y+3