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

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We have f(x) = 2x + 1. And g(x) = x + 2

Now gogog(x) = g(g(g(x))) = g(g(x+2)) = g(x +4) = x +6

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

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

=2*(4x +3) +1

= 8x + 6 +1

= 8x +7

So 8x +7 = x +6

=> 7x = -1

=> x = -1/7

**Therefore x = -1/7**

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.**