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

John B.

If possible, can you also help with number 408/19/20