Bunuel wrote:
Which of the following functions f defined for all numbers x has the property that f(–x) = –f(x) for all numbers x ?
A. \(f(x)= \frac{x^3}{(x^2 +1)}\)
B. \(f(x)= \frac{(x^2 - 1)}{(x^2 + 1)}\)
C. \(f(x)= x^2(x^2 - 1)\)
D. \(f(x) = x(x^3 - 1)\)
E. \(f(x) = x^2(x^3 - 1)\)
We'll show two approaches.
The first relies on number properties and is Logical.
Since an even power is always positive, then for any even powers f(x) = f(-x).
Similarly, for any odd powers f(x) = -f(-x).
So, we want there to be odd powers of x in our answer. But - if there are two such powers, the minus signs will cancel out.
So our correct answer must have an odd number of odd powers.
Looking at our answers, only (A) and (E) fit this criteria.
But, as (E) 'shifts' the value of x^3 by 1, it is not symmetrical with respect to 0 and cannot be our answer.
A less abstract answer is to try x = 1 in (E) and see that f(1) = 0 but f(-1) = -2.
The second answer ignores all the above logical considerations and just tries easy numbers.
This is an Alternative approach.
Let's start with x = 1 and x = -1
(A) gives f(1)=1/2 and f(-1) = -1/2. Maybe. Let's check the others.
(B) gives f(1)=0 and f(-1) = 0. Maybe. Let's check the others.
(C) gives f(1)=0 and f(-1) = 0. Maybe. Let's check the others.
(D) gives f(1)=0 and f(-1) = -2. No
(E) gives f(1)=0 and f(-1) = -2. No
To separate (A), (B), (C) we'll try x = 2 and x = -2.
(A) gives f(2)=8/5 and f(-2) = -8/5. Maybe. Let's check the others.
(B) gives f(2)=3/5 and f(-2) = 3/5. No
(C) gives f(2)=12 and f(-2) = 12. No
(A) is our answer.
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