Composition Of Functions Proof

Question

Hey, I need help with the following questions if anybody can help. Solutions + reason why would be very helpful too. Thanks in advance!

Composition of functions Proof Exercise

Given: 

1)

f(x) and g(x) are inverses of each other, and that h(x) relates to r(x) such that h(r)=g(x). 

Simplify the expression (PROVE)

 h(r(f)) 

2) Now given that you have theoretically proved this, make up actual functions for f, g, h, and r, to show that the proof works for a specific example and show all of your work. 

3) given f(2)=6 evaluate through a proof. 

g(f(h(r(6))): 

4) After theoretically showing the result above, again, make up functions f, g, h, and r to show that the proof actually works for a specific situation. Show all work. 

Answer 1

I presume that h(r) = g(x) means h(r(x)) = g(x)

Then, as g and f are inverses of one another, g(f(x)) = x

Thus, h(r(f(x))) = g(f(x)) = x

Let g(x) = x^6, h(x) = x^3, r(x) = x^2, and f(x) = x^(1/6)

f(g(x) = f(x^6) = (x^6)^(1/6) = x^1 = x

h(r(x)) = h(x^2) = (x^2)^3 = x^6

Then, h(r(f(x)) = h(r(x^1/6) = h(x^1/6)^2 = h(x^1/3) = (x^1/3)^3 = x

g(f(h(r(6)) = g(f(g(6)) = g(6) = g(f(2)) = 2, as g(f(x)) = x

As f(2) = 6, let's let f(x) = 3x, g(x) = 1/3 x, h(x) = 1/2x, and r(x) = 2/3 x

Note that 3(1/3 x) = 1/3(3x) = x and h(r(x)) = 1/2(2/3x) = 1/3x = g(x)

Then, g(f(h(r(6)) = g(f(h(2/3 * 6))) = g(f(h(4))) = g(f(1/2*4) = g(f(2)) = g(3*2) = g(6) = 1/3 * 6 = 2