6. Prove that f(x) = (3/4)x^4 + 2 and g(x) = (4sqrt(108x-216))/3 are inverses.

I know you solve it with composite functions f(g(x)) and g(f(x)), and I solved f(g(x)), but I can't seem to solve g(f(x)) to equal x, like it should. Can someone show me how to solve this? Thanks :)

The 4 in front of the sqrt is like the 4th root, like 3sqrt(... is the cubed root, etc.

You probably took a wrong turn in what was included inside the fourth root. Easy turn to miss. :-)

f(x)=(3/4)x^4

g(x)= ((108((3/4)x^4 + 2)-216)^(1/4))/3

= ((81x^4 + 216 -216)^1/4) /3

= ((81x^4)^1/4) /3

= 81^(1/4)x^(1/4) /3

= 3 x / 3

= x

Ohhhhh, I think I get it now. Thanks jim! :D :D

To prove that two functions, f(x) and g(x), are inverses of each other, you need to show that their compositions, f(g(x)) and g(f(x)), result in x.

Let's start by finding f(g(x)) and simplifying it:

f(g(x)) = f((4√(108x-216))/3)
= (3/4)((4√(108x-216))/3)^4 + 2
= (3/4)((4/3)^4)√(108x-216)^4 + 2
= (3/4)(256/81)(108x-216)^2 + 2
= (256/108)(108x-216)^2 + 2
= 2(108x-216)^2 + 2
= 2(108^2 * (x-2)^2) + 2
= 2 * 108^2 * (x-2)^2 + 2
= 2 * 11664 * (x-2)^2 + 2
= 23328 * (x-2)^2 + 2

Now, to solve for g(f(x)), equate it to x and solve for x:

g(f(x)) = x

Plug in f(x) = (3/4)x^4 + 2 into g(f(x)):

g(f(x)) = (4√(108((3/4)x^4 + 2) - 216))/3
= (4√(81x^4 + 216 - 216))/3
= (4√(81x^4))/3
= (4*9x^2)/3
= (12x^2)/3
= 4x^2

Now, set g(f(x)) = x and solve for x:

4x^2 = x

Rearrange the equation to standard form:

4x^2 - x = 0

Now, factor out x:

x(4x - 1) = 0

So, the solutions are x = 0 and 4x - 1 = 0, which gives x = 1/4.

Therefore, the functions f(x) and g(x) are inverses of each other because their compositions, f(g(x)) and g(f(x)), both result in x.