Relations and Functions - Algebra of real functions

Definition of Real Functions: A function $f: A \rightarrow B$ is called a real-valued function if $B$ is a subset of the set of real numbers $\mathbb{R}$. If $A$ is also a subset of $\mathbb{R}$, it is called a real function. Visually, the graph of such a function exists on a 2D coordinate plane where the horizontal axis represents the domain and the vertical axis represents the range.

Addition of Real Functions: For two real functions $f: X \rightarrow \mathbb{R}$ and $g: X \rightarrow \mathbb{R}$, their sum $(f+g): X \rightarrow \mathbb{R}$ is defined by $(f+g)(x) = f(x) + g(x)$. Graphically, this corresponds to 'pointwise addition' where at any given $x$, the $y$-coordinate of the new graph is the sum of the $y$-coordinates of $f$ and $g$.

Subtraction of Real Functions: The difference $(f-g): X \rightarrow \mathbb{R}$ is defined by $(f-g)(x) = f(x) - g(x)$. If we visualize the graphs of $f$ and $g$, the resulting graph of $f-g$ represents the vertical distance between the two curves at each $x$. If the curves intersect, the value of the difference function is zero, crossing the $x$-axis.

Multiplication by a Scalar: For a real function $f$ and a scalar $\alpha$, the function $(\alpha f)(x) = \alpha \cdot f(x)$. Visually, if $\alpha > 1$, the graph undergoes a vertical stretch; if $0 < \alpha < 1$, it undergoes a vertical compression. If $\alpha$ is negative, the graph is reflected across the $x$-axis.

Multiplication of Real Functions: The pointwise product $(fg): X \rightarrow \mathbb{R}$ is defined by $(fg)(x) = f(x)g(x)$. An important visual property is that if either $f(x)=0$ or $g(x)=0$ at a point $x$, the product graph will have an $x$-intercept (root) at that point.

Quotient of Real Functions: The quotient of two functions is defined as $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$. On a graph, points where $g(x) = 0$ but $f(x) \neq 0$ typically result in vertical asymptotes, where the curve approaches infinity as it nears that $x$-value.

Domain Intersection Rule: When performing algebraic operations on two functions $f$ and $g$, the domain of the resulting function (sum, difference, or product) is the intersection of their individual domains: $D_f \cap D_g$. For the quotient $f/g$, we must further exclude all points where the denominator $g(x)$ equals zero.