Algebra of Real Functions

Algebra of Real Functions

When you have two real functions — that is, functions whose domain and codomain are subsets of the real numbers — you can combine them using the same arithmetic operations you use with ordinary numbers: addition, subtraction, multiplication, and division. The key idea is that these operations are performed pointwise: for each input $x$, you apply the operation to the outputs $f(x)$ and $g(x)$.

For all the definitions that follow, we assume $f: X \to \mathbb{R}$ and $g: X \to \mathbb{R}$ are two real functions, where $X \subset \mathbb{R}$.

Addition of Two Real Functions

The sum of $f$ and $g$ is a new function, written $f + g$, defined by

The domain of $f+g$ is the same as the common domain $X$ of $f$ and $g$. At every point $x$, you simply add the two function values.

Subtraction of a Real Function from Another

The difference of $f$ and $g$, written $f - g$, is defined by

Again, the domain is $X$, and the operation is pointwise.

Multiplication by a Scalar

Let $\alpha$ be a scalar — here, a scalar means a real number. The product of $\alpha$ and $f$, written $\alpha f$, is a function defined by

This scales every output of $f$ by the constant factor $\alpha$. The domain remains $X$.

Multiplication of Two Real Functions

The product of $f$ and $g$, written $fg$, is defined by

This is called pointwise multiplication because you multiply the values at each point individually. The domain is $X$.

Quotient of Two Real Functions

The quotient of $f$ by $g$, written $\frac{f}{g}$, is defined by

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad \text{for all } x \in X \text{ such that } g(x) \neq 0.$ …