Given f(x)=2x+1 and g(x)=x+2 solve the equation gogog(x)=fofof(x)

We'll determine (gogog)(x) =
g(g(g(x)))

g(g(g(x))) = g(g(x)) +
2

g(g(g(x))) = [g(x) + 2] +
2

g(g(g(x))) = (x + 2 + 2) +
2

g(g(g(x))) = x + 6

We'll
determine (fofof)(x) = f(f(f(x)))

f(f(f(x))) = 2f(f(x)) +
1

f(f(f(x))) = 2[2f(x) + 1] +
1

f(f(f(x))) = 4f(x) + 2 +
1

f(f(f(x))) = 4(2x + 1) +
3

f(f(f(x))) = 8x + 4 +
3

f(f(f(x))) = 8x + 7

We'll
solve the equation:

x + 6 = 8x +
7

We'll subtract 8x + 7 both
sides:

x - 8x + 6 - 7 = 0

-7x
- 1 = 0

We'll add 1:

-7x =
1

We'll divide by -7 both
sides:

x =
-1/7

The solution of the equation g(g(g(x)))
= f(f(f(x))) is x = -1/7.