Learning objective
There are two special types of polynomial functions based on the degree of each term of the polynomial.
If the degrees of some terms in a function are even and the degrees of other terms in a function are odd, then the function is neither even nor odd. This means its graph will not be symmetric to the y-axis nor the origin. When we substitute -x into the function and simplify, some signs will change, and others will not.
Determine if the following functions are even, odd, or neither.
f\left(x\right)=-6x^5+15x^3-9x
p\left(x\right)=\sqrt{6}x^4-\sqrt{11}x^2-\sqrt{5}
A function is even if f\left(-x\right)=f\left(x\right). In other words, if we substitute -x into the function and simplify, the result is the same as the original function. Even functions are symmetric about the y-axis, and the degree of each term is even or 0.
A function is odd if f\left(-x\right)=-f\left(x\right). In other words, if we substitute -x into the function and simplify, the signs of all the terms should be opposite of the terms in the original function. Odd functions are symmetric about the origin, and the degree of each term is odd.