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If f(x) is a function, what is the value of f(2)?

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Re: If f(x) is a function, what is the value of f(2)? [#permalink]
Bunuel wrote:
If f(x) is a function, what is the value of f(2)?

(1) $$(f(x))^2 = 2f(x) − 1$$

(2) $$f(4x) = 8x^2 − 1$$

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Since we have 1 variable (f(2)) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
$$(f(x))^2 = 2f(x) - 1$$
$$⇔ (f(x))^2 - 2f(x) + 1 = 0$$
$$⇔ (f(x)-1)^2 = 0$$
$$⇔ f(x) = 1$$.
Thus, $$f(2) = 1$$.
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
When we plug-in $$\frac{1}{2}$$ into $$x$$, we have $$f(2) = 8(\frac{1}{2})^2-1 = 1$$.
Since condition 2) yields a unique solution, it is sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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Re: If f(x) is a function, what is the value of f(2)? [#permalink]
Bunuel wrote:
If f(x) is a function, what is the value of f(2)?

(1) $$(f(x))^2 = 2f(x) − 1$$

(2) $$f(4x) = 8x^2 − 1$$

Given: f(x) is a function

Asked: What is the value of f(2)?

(1) $$(f(x))^2 = 2f(x) − 1$$
(f(2))^2 = 2f(2) − 1
Let f(2) be x
x^2 - 2x + 1 = 0
(x -1)^2 = 0
x = 1
f(2) = 1
SUFFICIENT

(2) $$f(4x) = 8x^2 − 1$$
f(4x) = 8x^2 − 1
Let x = 1/2
f(2) = 8(1/2)^2 - 1 = 8/4 - 1= 2-1 = 1
f(2) = 1
SUFFICIENT

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Re: If f(x) is a function, what is the value of f(2)? [#permalink]
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Re: If f(x) is a function, what is the value of f(2)? [#permalink]
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