If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does : GMAT Problem Solving (PS)
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If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does

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If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does [#permalink]

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10 Jul 2013, 11:52
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If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does f(g(x)) = g(f(x))?

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

Last edited by Bunuel on 10 Jul 2013, 12:02, edited 1 time in total.
Edited the question.
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Re: If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does [#permalink]

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10 Jul 2013, 12:10
josemnz83 wrote:
If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does f(g(x)) = g(f(x))?

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

Given: $$f(x) = x^3 + 1$$ and $$g(x) = 2x$$.

LHS: $$f(g(x))=f(2x)=(2x)^3+1=8x^3+1$$

RHS: $$g(f(x))=g(x^3+1)=2(x^3+1)=2x^3+2$$

So, we have that $$8x^3+1=2x^3+2$$ --> $$x^3=\frac{1}{6}$$ --> $$x=\frac{1}{\sqrt[3]{6}}$$.

Hope it's clear.
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Manager
Joined: 30 Jun 2012
Posts: 84
Location: United States
GMAT 1: 510 Q34 V28
GMAT 2: 580 Q35 V35
GMAT 3: 640 Q34 V44
GMAT 4: 690 Q43 V42
GPA: 3.61
WE: Education (Education)
Followers: 13

Kudos [?]: 123 [0], given: 16

Re: If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does [#permalink]

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10 Jul 2013, 12:18
On the RHS, g(x)=2x. You wrote 2(x^3+1). What happened to the x in the 2x? That is where I get lost.
Thanks!
Bunuel wrote:
josemnz83 wrote:
If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does f(g(x)) = g(f(x))?

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

Given: $$f(x) = x^3 + 1$$ and $$g(x) = 2x$$.

LHS: $$f(g(x))=f(2x)=(2x)^3+1=8x^3+1$$

RHS: $$g(f(x))=g(x^3+1)=2(x^3+1)=2x^3+2$$

So, we have that $$8x^3+1=2x^3+2$$ --> $$x^3=\frac{1}{6}$$ --> $$x=\frac{1}{\sqrt[3]{6}}$$.

Hope it's clear.
Math Expert
Joined: 02 Sep 2009
Posts: 36566
Followers: 7079

Kudos [?]: 93179 [1] , given: 10553

Re: If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does [#permalink]

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10 Jul 2013, 12:23
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Expert's post
josemnz83 wrote:
On the RHS, g(x)=2x. You wrote 2(x^3+1). What happened to the x in the 2x? That is where I get lost.
Thanks!
Bunuel wrote:
josemnz83 wrote:
If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does f(g(x)) = g(f(x))?

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

Given: $$f(x) = x^3 + 1$$ and $$g(x) = 2x$$.

LHS: $$f(g(x))=f(2x)=(2x)^3+1=8x^3+1$$

RHS: $$g(f(x))=g(x^3+1)=2(x^3+1)=2x^3+2$$

So, we have that $$8x^3+1=2x^3+2$$ --> $$x^3=\frac{1}{6}$$ --> $$x=\frac{1}{\sqrt[3]{6}}$$.

Hope it's clear.

This might help: $$g(n) = 2n$$ --> $$g(x^3+1)=2(x^3+1)$$ ($$n=x^3+1$$).
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Re: If f(x) = x^3 + 1, and g(x) = 2x, for what value of x does   [#permalink] 10 Jul 2013, 12:23
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