Question

Out of the following functions, which would be odd?

`g(x)=-1/2(x+5)^3+2`

`g(x)=2(x-3)^2-8`

`g(x)=-4/x`

`g(x)=sqrt(2x+6)-5`

`g(x)=||-x+5||`

Answer 1

`g(x)=-4/x` is the only odd function given.

Explanation:

I wrote a tutorial for even and odd functions in which all techniques used here are explained. The techniques described may tell you at a glance whether a function is even or odd, but may not be accepted as reasoning by a teacher. As such, alternate reasoning is also provided in some cases.

• `g(x)=-1/2(x+5)^3+2` - Not odd

If we expanded the cubed binomial, we would have a polynomial with both even and odd exponents. This would give us a sum of terms which are even and odd functions, meaning `g(x)` is neither even nor odd.

Alternately, consider the counterexample `g(-5) != -g(5)`

• `g(x)=2(x-3)^2-8` - Not odd

Similar to the above, expanding the squared binomial would give us at least an `x^2` term and an `x` term, meaning the whole polynomial is neither even nor odd.

Alternately, consider the counterexample `g(-3) != -g(3)`

• `g(x)=-4/x` - Odd

This is the product of the constant `-4` (even) and the exponential `x^-1` (odd), and the product of an even and an odd function is odd.

Alternately, using the definition of an odd function:

`g(-x) = -4/(-x) = -(-4/x) = -g(x)`

• `g(x)=sqrt(2x+6)-5` - Not odd

Consider the counterexample `g(-3) != -g(3)`

• `g(x)=||-x+5||` - Not odd

Consider the counterexample `g(-5) != -g(5)`