The exercise is about polynomials. A polynomial is a function composed by the sum of powers of a variables, with some coefficients.
Now, a polynomial can contain all powers of a variable, with odd and even exponents, for example
[tex] x^4+3x^3-5x^2-6x+8 [/tex] contains (coefficients apart) [tex] x^4, x^2 \text{ and } x^0=1 [/tex], which are odd powers of [tex] x [/tex], as well as [tex] x^3\text{ and } x [/tex], which are odd powers.
Now, assume that we have instead a polynomial consisting of only odd powers. For example,
[tex] 6x^7+4x^5-x [/tex]
The question is: is this polynomial an odd function? Well, a function is said to be odd if [tex] f(-x) = -f(x) [/tex]
In other words, if you give an odd function the opposite of the input, you get the opposite of the output.
And odd powers behave exactly like this: you have
[tex] (-x)^3 = -x^3,\quad (-x)^5 = -x^5,\quad (-x)^7=-x^7[/tex]
and so on. If we consider the same example as before, you have
[tex] f(x) = 6x^7+4x^5-x \implies f(-x) = 6(-x)^7+4(-x)^5-(-x) = -6x^7-4x^5+x = -f(x)[/tex]
So yes, a function where only odd powers of [tex] x [/tex] appear is an odd function.
