Section Summary

The central concept in this section is the notion of a function as an object. In particular we consider classes of functions described by functional equations.

• A function `f` is said to be additive if

  ${\displaystyle f(a+b)=f\left(a\right)+f\left(b\right)}$.

  The collection of additive functions includes those defined by $f\left(x\right)=cx\text{,}$ where `c` is any constant.

• A function `f` is said to be multiplicative if

  ${\displaystyle f(ab)=f\left(a\right)f\left(b\right)}$.

  The collection of multiplicative functions includes those defined by $f\left(x\right)=x^k$, where `k` is any constant.

• A function `f` is said to be odd if

  $f(-x)=-f\left(x\right)$.

  The collection of odd functions includes all power functions of the form $f\left(x\right)=x^n$, where `n` is odd.

• A function `f` is said to be even if

  $f(-x)=f\left(x\right)$.

  The collection of even functions includes all power functions of the form $f\left(x\right)=x^n$, where `n` is even.