Free Practice Question

PS · Problem Solving

705-805

Functions

If function $f(x)$ satisfies $f(x) = f(x^2)$ for all $x$ , which of the following must be true?

Answer Choices

Correct answer marked below

A

f(4) = f(2)f(2)

f(16) - f(-2) = 0

Correct

C

f(-2) + f(4) = 0

D

f(3) = 3f(3)

E

f(0) = 0

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Explanation

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Official Explanation

Official

We are told that a particular function

f(x)

has the property

f(x) = f(x^2)

for all values of

x

. Note that we don't know the specific form of the function; we only know this one property.

For example, for this function,

f(3)=f(3^2)

or

f(3)=f(9)

; similarly,

f(9)=f(81)

, so

f(3)=f(9)=f(81)=\ldots

.

Now, the question asks: which of the following MUST be true?

A.

f(4)=f(2)f(2)

: we know that

f(2)=f(4)

, but it's not necessarily true that

f(2)=f(2)f(2)

(this will only be true if

f(2)=1

or

f(2)=0

, but as we don't know the actual function, we cannot say for sure).

B.

f(16) - f(-2) = 0

: since

f(-2)=f(4) =f(16)=...

, we have that

f(16)-f(-2)=f(16)-f(16)=0

, so this option is always true.

C.

f(-2) + f(4) = 0

:

f(-2)=f(4)

, but it's not necessarily true that

f(4) + f(4)=2f(4)=0

(this will only be true if

f(4)=0

, but again we don't know that for sure).

D.

f(3)=3f(3)

: is

3f(3)-f(3)=0

? is

2f(3)=0

? is

f(3)=0

? Since we don't know the actual function, we cannot say for sure.…

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If function $f(x)$ satisfies $f(x) = f(x^2)$ for all $x$ , which of the following must be true?

Answer Choices

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Official Explanation

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What it tests

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If function f(x)f(x)f(x) satisfies f(x)=f(x2)f(x) = f(x^2)f(x)=f(x2) for all xxx, which of the following must be true?

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Official Explanation

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What it tests

Common trap

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If function $f(x)$ satisfies $f(x) = f(x^2)$ for all $x$ , which of the following must be true?