Vincen wrote: ↑

Tue Aug 18, 2020 7:48 am

Which of the following functions f has the property that \(f(-x)=-f(x)\) for all numbers \(x?\)

A. \(\dfrac{x^4-3}{x^6+2}\)

B. \(-\dfrac{x^2}{(x-1)^2}\)

C. \(\dfrac{x^3-7}{(x-2)^2}\)

D. \(\dfrac{x^4-x}{x^2+2}\)

E. \(\dfrac{x^5}{x^2+4}\)

Answer: E

Solution:

A function that has the property f(-x) = -f(x) is called an odd function. In other words, the question asks us to determine the function in the given choices that is odd. Of course, there are even functions also. To be an even function, the function should have the property f(-x) = f(x).

A function is a polynomial function when all the terms are of the form ax^n, where n is a nonnegative integer. For example, x^2 + 5, x^3 - 4x and x^4 + x^3 - 1 are polynomial functions and so are the functions that are either the numerators or the denominators of the given choices. The quotient of two polynomial functions is called a rational function. For example, all the given choices are rational functions.

An easy way to determine whether a polynomial function is odd or even is by its terms:

1) If all of its terms have odd exponents, then it’s odd. For example, x^3 - 4x = x^3 - 4x^1.

2) If all of its terms have even exponents, then it’s even. For example, x^2 + 5 = x^2 + 5x^0. (Recall that 0 is an even number and x^0 = 1, so 5 = 5x^0.)

3) If some of its terms have odd exponents and some have even exponents, then it’s neither odd nor even. For example, x^4 + x^3 - 1.

Now an easy way to determine whether a rational function (since all our choices are rational functions) is odd or even is:

1) If the numerator and denominator polynomial functions are both even or both odd, then the rational function is even.

2) If the numerator polynomial function is odd and the denominator polynomial function is even, and vice versa, then the rational function is odd.

3) If the numerator (or denominator) polynomial function is neither, then regardless of what the other polynomial function is, the rational function is neither.

If you are puzzled on why (odd function)/(odd function) is an even function, just think in terms of subtraction of two odd numbers, odd - odd = even. After all, when we divide two terms with like base, we subtract their exponents. For example, x^5/x^3 = x^2. In this case, we see that both x^5 and x^3 are odd functions, but the quotient, x^2 is an even function. The idea expands to, for example, functions such as (x^3 - 4x)/(x^2 + 5), despite the fact that we can’t actually simplify it. As we can see, (x^3 - 4x)/(x^2 + 5) is an odd function since (odd function)/(even function) = odd function (again just think it as odd number - even number = odd number).

Now, let’s go over the given choices using what we’ve mentioned above (remember that we need to determine the one that is an odd function).

A) (x^4 - 3)/(x^6 + 2) = even/even = even

B) -x^2/(x - 1)^2 = -x^2/(x^2 - 2x + 1) = even/neither = neither (Notice that we need to expand the denominator first so that we can see the exponents on each term).

C) (x^3 - 7)/(x - 2)^2 = (x^3 - 7)/(x^2 - 4x + 4) = neither/neither = neither

D) (x^4 - x)/(x^2 + 2) = neither/even = neither

E) x^5/(x^2 + 4) = odd/even = odd

Answer: E