Question

Which of these is an odd function?

Answer 1

Answer:

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

Step-by-step 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!