For which of the following functions does f(x)=f(x) for all real num

E = -X^5/(X^2+1)...Both the sides

Answer = E. x^5/(x^2+1)

Change of sign happens does NOT happen when:

1. All powers are even

2. All powers are odd

If you observe the OA, only option E has the combination of odd power in numerator & even in denominator

\(f(-x) = \frac{(-x)^5}{(-x)^2+1} = \frac{-x^5}{x^2+1}\)

\(-f(x) = \frac{-x^5}{x^2+1}\)

Answer = E

Use x = 2 as your guide to test the answer choices.

For choice A, f(−x) will be the same as f(x), because the even exponents ensure that whether you're taking 2 or -2 to the 8th or 4th powers the results will always be the same. Therefore, since f(−x)=f(x), then this doesn't satisfy the prompt "does f(−x)=−f(x).

Mathematically, using x = 2 you'll have f(−x)=(−2)8−(−2)4=256−16=230. Meanwhile, −f(x)=−(28−24)=−(256−16)=−230, so you don't have a match.

For choice B, the logic and math are pretty similar. Since both exponents are even, whether you're using a positive or negative value as your input, the results will be the same either way for f(2) and f(−2). So your positive/negative eyeball test should tell you that this one won't work either.

Mathematically, using x = 2, you'll have: f(−x)=(−2)2−(−2)6=4−64=−60. And −f(x)=−(22−26)=−(4−64)=−(−60)=60 They're not the same, so choice B doesn't hold.

Choice C is similar, in that x^5/x^7 is going to net out to 1x/^2. A negative input like 2 will still yield a positive output (1/4), whereas −f(x) is going to yield a negative number (f(x)=1/4, so −f(x)=−1/4).

By now you should see that choice D will be the same - with two odd exponents, the negative input will yield a negative divided by a negative, and a positive number overall. While taking −f(x) will yield a negative.

Mathematically, that's f(−x)=(−2)^9/(−2)^5+1=−29−31, which is a positive number, whereas −f(x)=−(2925+1), where the answer will clearly be negative.

Only choice E mixes an odd exponent with an even exponent, a necessary pairing to satisfy the question. Mathematically with 2, you'd have:

f(−x)=(−2)^5/(−2^)2+1=−32/5, and

−f(x)=−2^5/2^2+1=−32/5, for a match.

ANSWER: E

A. \(f(-x) = (-x)^8 - (-x)^4\) is not equal to \(-f(x) = -(x^8 - x^4).\)

B. option B is the same as option A− \(f(-x) = (-x)^2 - (-x)^6\) is not equal to \(-f(x) = -(x^2 - x^6).\)

C. \(f(-x) = \frac{(-x)^5}{(-x)^7}\) is not equal to \(-f(x) = \frac{-(x^5)}{(x^7)}\)\(. f(-x)\) yields a + answer, whilst \(-f(x)\) yields a − answer.

D. \(f(-x) = \frac{(-x)^9}{((-x)^5+1)}\) is not equal as \(-f(x) = \frac{(-1)*x^9}{(x^5+1)}\).

E. \(f(-x) = \frac{(-x)^5}{((-x)^2+1)}\) is equal to \(-f(x) = \frac{(-1)*x^5}{(x^2+1)}\), because \((-x)^5\) is equal to \((-1)*x^5\) and \(((-x)^2+1) \)is the same as \((x^2+1)\).

Let x = 1

A: f(-1) -> 0
B: f(-1) -> 0
C: f(-1) -> 1, and f(1) = 1
D: f(-1) -> -1/0
E: f(-1) -> -1/2, and f(1) = 1/2

Answer: E

I misread the answer D as if the +1 were adding the exponent, in which case it would become x^6 and D would be the answer.

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