Algebra, Ch. 5, p118, compound functions

[jtoallen1]

I'm having trouble understanding the final example in the compound functions section of this chapter. Can you please help?

We are supposed to find x, when:

f(g(x)) = g(f(x))

given that

f(x) = X^3 + 1

g(x) = 2x

I am having trouble getting from this step:

f(2x) = g(x^3 + 1)

To this step:

(2x)^3 +1 = 2(x^3 + 1)


Can you please show the steps involved and common mistakes.Thanks!

[n00bpron00bpron00b]

f(g(x)) = g(f(x)) ---- (1)

given that

f(x) = X^3 + 1 ----(2)
g(x) = 2x ---- (3)


Okay, f(g(x)) = g(f(x))

Lets consider the LHS of equation (1) -

f(g(x)) =

substituting the value of g(x) here
f (2x)

now we need to find the value of f(2x), substituting (2x) in f(x) equation (i.e equation 2)

f(2x) = (2x)^3 + 1
f(2x) = 8x^3 + 1 ------- (4)

Now RHS of equation (1) -

g(f(x)) =

substituting the value of f(x) here

g(x^3+1)

now we need to find the value of g(x^3+1), substituting (x^3+1) in the g(x) equation (i.e in equation 3)

g(x^3+1) = 2(x^3+1) ---- (5)

given that f(g(x)) = g(f(x))

equate equations (4) & (5)

(8x^3) + 1 = (2x^3) + 2

(8x^3) - (2x^3) = 2 -1
6x^3 = 1
x^3 = 1/6
x = cube root of (1/6)

[jtoallen1]

Thank you, thank you! that clears it up for me. the book didn't outline all the steps and your work finally led me to the point "where it clicked" and made sense!

[n00bpron00bpron00b]

no problem :)

[tommywallach]

Noob is the greatest!