Official Solution:For which of the following functions \(f(x)\) is the relation \(f(f(x)) = f(f(f(f(x))))\) NOT true for at least some values of \(x\) not equal to zero?A. \(f(x) = -|x|\)

B. \(f(x) = 2 - x\)

C. \(f(x) = 3x\)

D. \(f(x) = \frac{4}{x}\)

E. \(f(x) = 5\)

We are asked which function does NOT obey the rule \(f(f(x)) = f(f(f(f(x))))\). This rule looks intimidating, but all it means on the left side is that we put some number \(x\) into the function, get the output, and then put that output back into the function and see what we get out. We do the same thing (feeding the function its own output) two more times on the right side, then compare the two sides.

Now let's look at the functions.

(A) \(f(x) = -|x|\)

This function takes the absolute value of \(x\), then puts a negative sign on. For instance, if \(x = 7\), then \(f(x) = -|7| = -7\). Likewise, if \(x = -8\), then \(f(x) = -|-8| = -8\). In words, \(f(x)\) turns any number negative (it's the "negative absolute value"). Applying this process twice gives you the same number as applying it 4 times. INCORRECT.

(B) \(f(x) = 2 - x\)

Let's see what happens when we try to calculate \(f(f(x))\). Work your way from the inside out:

\(f(f(x)) = f(2 - x) = 2 - (2 - x) = 2 - 2 + x = x\). In other words, \(f(f(x))\) just gives us \(x\) back. Therefore, applying two MORE \(f\)'s to get \(f(f(f(f(x))))\) will give us \(x\) again as well. INCORRECT.

(C) \(f(x) = 3x\)

If \(f(x) = 3x\), then \(f(f(x)) = f(3x) = 3(3x) = 9x\). Applying two MORE \(f\)'s to get \(f(f(f(f(x))))\) will give us \(3(3(9x) = 81x\). \(9x\) does NOT equal \(81x\) for any nonzero \(x\), in fact. CORRECT.

(D) \(f(x) = \frac{4}{x}\)

We should finish out the list, just to make sure.

\(f(f(x)) = f(\frac{4}{x}) = \frac{4}{\frac{4}{x}} = x\). As with the function in (B), this function brings us back to \(x\) if we apply it twice. Thus, if we apply it 4 times, we also get back to \(x\). INCORRECT.

(E) \(f(x) = 5\)

\(f(f(x)) = f(5) = 5\). This function may seem tricky, but it's actually very simple: it gives you back a 5 no matter what you feed into it. If you give it a 5, in particular, you still get a 5 back, no matter how many times you go through that cycle. INCORRECT.

Answer: C

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