Sets and Functions - Sum, Difference, Product and Quotients of Functions

Algebra of Real Functions: This involves performing arithmetic operations on two functions $f$ and $g$ that map from a domain $D$ to the set of real numbers $\mathbb{R}$. The resulting function is also a real-valued function defined over a specific common domain.

The Domain of Combined Functions: For the sum, difference, and product of functions $f$ and $g$, the domain is the intersection of their individual domains, denoted as $D_f \cap D_g$. Visually, this is represented on a number line as the overlapping segment where both functions are simultaneously defined.

Addition of Functions: The sum $(f + g)(x)$ is obtained by adding the outputs of $f(x)$ and $g(x)$ for every $x$ in the intersection of their domains. Graphically, the $y$-coordinate of the sum function at any point $x$ is the vertical sum of the individual $y$-coordinates of $f$ and $g$.

Subtraction of Functions: The difference $(f - g)(x)$ is calculated as $f(x) - g(x)$. On a graph, this represents the vertical distance between the curves of $f$ and $g$ at a specific $x$ value, provided that $x$ exists in $D_f \cap D_g$.

Multiplication of Functions: The product $(f \cdot g)(x)$ is the point-wise product $f(x) \cdot g(x)$. If one function is a constant $c$, the operation $(cf)(x) = c \cdot f(x)$ represents a vertical stretch or compression of the graph of $f$ by a factor of $c$.

Quotient of Functions: The quotient $(\frac{f}{g})(x)$ is defined as $\frac{f(x)}{g(x)}$. Crucially, the domain for the quotient is $D_f \cap D_g$ excluding any $x$ such that $g(x) = 0$. Visually, these excluded points often appear as vertical asymptotes or 'holes' on the graph of the resulting function.

Scalar Multiplication: Multiplying a function by a scalar $c$ creates a new function $(cf)(x) = c f(x)$. If $c > 1$, the graph is vertically stretched; if $0 < c < 1$, the graph is vertically compressed; if $c$ is negative, the graph is reflected across the x-axis.