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

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### 2 Answers

It is given that f(x)=2x+1 and g(x)=x+2. We need to solve the equation gogog(x)=fofof(x)

gogog(x)

=> gog(x + 2)

=> g(x + 2 + 2)

=> g(x + 4)

=> x + 6

fofof(x)

=> fof(2x + 1)

=> f(4x + 2 + 1)

=> f(4x + 3)

=> 8x + 7

Equating x + 6 = 8x + 7

=> 7x = -1

=> x = -1/7

**The required solution of the equation gogog(x)=fofof(x) is x = -1/7**

We'll write the rule of composition:

(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 (1)

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 (2)

We'll equate (1) and (2):

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.**