Bunuel wrote:

If g is a function defined for all x by \(g(x) = \frac{x^4}{16}\), then what is the value of \(g(2x)\) in terms of \(g(x)\)?

(A) \(\frac{g(x)}{16}\)

(B) \(\frac{g(x)}{4}\)

(C) \(4g(x)\)

(D) \(8g(x)\)

(E) \(16g(x)\)

Let’s begin by calculating g(2x):

g(2x) = (2x)^4/16 = (16x^4)/16 = x^4

Now, since g(x) = (x^4)/16, we have:

g(2x) = x^4 = (16x^4)/16 = 16[(x^4/16)] = 16g(x)

Alternate Solution:

When x = 1, g(2x) = g(2) = (2^4)/16 = 1. We will test each answer choice to see which ones produce 1 when x = 1. Note that g(1) = 1^4/16 = 1/16.

Answer Choice A: g(x)/16

When x = 1, we have g(1)/16 = (1/16)/16 = 1/16^2 ≠ 1.

Answer Choice B: g(x)/4

When x = 1, we have g(1)/4 = (1/16)/4 = 1/64 ≠ 1.

Answer Choice C: 4g(x)

When x = 1, we have 4g(x) = 4(1/16) = 4/16 = 1/4 ≠ 1.

Answer Choice D: 8g(x)

When x = 1, we have 8g(x) = 8(1/16) = 8/16 = 1/2 ≠ 1.

Answer Choice E: 16g(x)

When x = 1, we have 16g(x) = 16(1/16) = 16/16 = 1.

Answer: E

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