Question

Which of these is an odd function?

Question 11 options:

f(x) =3x2+7

f(x) =2x3+x

f(x) =5x2−6

f(x) =4x3+2x2

Answer 1

`f(x)=2x^3+x`

Explanation:

An odd function f is one where
`f(-x)=-f(x)`. We can interpret this is meaning: reflecting the graph horizontally
`f(-x)` has the same effect as reflecting it vertically
`-f(x)` . The only graphs that meet this requirement are ones with reflectional symmetry across the line y = x, so we can immediately elimate the functions
`f(x)=3x^2+7` and
`f(x)=5x^2-6`, which have horizontal symmetry, but not the kind of symmetry we're looking for.

That leaves us with
`f(x)=2x^3+x` and
`f(x)=4x^3+2x^2`. One feature of exponents we can utilize is that -1 to an even power is 1, while -1 to an odd power is -1. If we want
`f(-x)=-f(x)`, we need to flip the signs of all the coefficients, and we can only do that if all the powers of x are odd. The only function with only odd powers of x is
`f(x)=2x^3+x`, and plugging in -x to the function reveals that

`f(-x)=2(-x)^3+(-x)\\f(-x)=-2x^3-x=-(2x^3+x)=-f(x)`

So
`f(x)=2x^3+x` the only odd function on the list!