SOLUTION: if f(g(x)) = 6x^3, what are possibilities for f,g? a) f(x) = 6x, g(x) = x^2 b) f(x) = x^2, g(x) = 6x c) f(x) = 2x, g(x) = 3x^3 d) f(x) = 3x^3, g(x) = 2x ** Please help me

Sometimes multiple choice questions are best solved by trying out each
choice to find out which one works:

Notice what is different about the notation f(x) and the notation f(g(x))

If you replace the x in f(x), by g(x), you get f(g(x))

So what we do with the right sides of f(x) and g(x) is the same. We replace
x in the right side of the f(x) equation by the ENTIRE RIGHT SIDE of the
g(x) equation.

 

if f(g(x)) = 6x³, what are possibilities for f,g?

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Let's see if (a) is the correct answer:

a) f(x) = 6x, g(x) = x²

Let's substitute the right side of the g(x) equation for x
in the f(x) equation:

f(g(x)) = 6(x²) = 6x² 

So (a) is not correct.

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Let's see if (b) is the correct answer:

b) f(x) = x², g(x) = 6x

Let's substitute the right side of the g(x) equation for x
in the f(x) equation:

f(g(x)) = (6x)² = 36x²

So (b) is not correct.

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Let's see if (c) is the correct answer:

c) f(x) = 2x, g(x) = 3x³

Let's substitute the right side of the g(x) equation for x
in the f(x) equation:

f(g(x)) = 2(3x³) = 6x³

So (c) is the correct choice.  

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We have the answer but let's follow through and see why (d) isn't correct:

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Let's see if (d) is the correct answer:

d) f(x) = 3x^3, g(x) = 2x

Let's substitute the right side of the g(x) equation for x
in the f(x) equation:

f(g(x)) = 3(2x)³ = 3·2³x³ = 3·8x³ = 24x³

So (d) is not correct.

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Answer (c)


Edwin