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# For which of the following functions f(x) is the relation f(f(x)) =

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Math Expert
Joined: 02 Sep 2009
Posts: 51098
For which of the following functions f(x) is the relation f(f(x)) =  [#permalink]

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11 Nov 2014, 07:58
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15
00:00

Difficulty:

55% (hard)

Question Stats:

64% (02:03) correct 36% (01:35) wrong based on 261 sessions

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Tough and Tricky questions: Functions.

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. $$f(x) = 5$$

Kudos for a correct solution.

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Re: For which of the following functions f(x) is the relation f(f(x)) =  [#permalink]

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12 Nov 2014, 00:24
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2
Actually its asking for what options given in the OA, $$f(f(x))\neq{f(f(f(f(x))))}$$

Option A

f(x) = -|x|

f(1) = -1

f(f(1)) = -| f(1) | = - |-1| = -1

This always gives an -ve result of the any +ve / -ve value of x; so IGNORE

Option B

f(x) = 2 - x

f(1) = 2-1 = 1

f(f(1)) = 2 - f(1) = 2 -1 = 1

Returning same value, IGNORE

Option C

f(x) = 3x

f(1) = 3*1 = 3

f(f(1)) = 3 * f(1) = 3*3 = 9 .....Correct Answer

Option D

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

$$f(1) = \frac{4}{1} = 4$$

$$f(f(1)) = \frac{4}{f(1)} = \frac{4}{4} = 1$$

$$f(f(f(1))) = \frac{4}{f(f(1))} = \frac{4}{1} = 4$$

$$f(f(f(f(1)))) = \frac{4}{f(f(f(1)))}= \frac{4}{4} = 1$$

Returning same value, IGNORE

Option E

f(x) = 5

There is no variable "x" on the RHS. So value remains the same

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Joined: 21 Jul 2014
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Re: For which of the following functions f(x) is the relation f(f(x)) =  [#permalink]

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11 Nov 2014, 12:33
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Bunuel wrote:

Tough and Tricky questions: Functions.

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. $$f(x) = 5$$

Kudos for a correct solution.

For this problem, you might be inclined to try each function to see how they are working. If you just look through the answer choices, though, 3x looks like an easy function to try since it is simple and should stay increasing or decreasing depending on your initial value of x regardless of how many times it is looped.

Since we only need to find one instance where f(f(x) does not equal f(f(f(f(x)))), we can just plug in a number for x and try it.

I chose x = 1 for simplicity's sake. this yields f(f(x))= 9 and f(f(f(f(x)))) = 81, giving us a quick and easy example of an instance where choice C does not hold true for the given requirement.

So choice C is the answer.
Math Expert
Joined: 02 Sep 2009
Posts: 51098
Re: For which of the following functions f(x) is the relation f(f(x)) =  [#permalink]

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12 Nov 2014, 03:01
2
1
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.

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Re: For which of the following functions f(x) is the relation f(f(x)) =  [#permalink]

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04 Aug 2018, 20:03
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Re: For which of the following functions f(x) is the relation f(f(x)) = &nbs [#permalink] 04 Aug 2018, 20:03
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