Understanding the Algebra of Functions

The algebra of functions addresses the arithmetic operations definable for real functions, specifically the precise rules for addition, subtraction, multiplication, and division of two or more functions with overlapping domains. These operations form the algebraic backbone for manipulating functions, essential in advanced mathematics and competitive examinations.

Addition and Subtraction of Real Functions: Pointwise Formalism and Domain

Given two real functions $f$ and $g$ with respective domains $D_f$ and $D_g$, the sum function $(f+g)$ is defined at all $x$ common to both domains (i.e., $x\in D_f\cap D_g$). For such $x$, the operation is given by

$$(f+g)(x) = f(x) + g(x).$$

Similarly, the difference function $(f-g)$ is defined at all $x\in D_f\cap D_g$ as

$$(f-g)(x) = f(x) - g(x).$$

The domain of both $(f+g)$ and $(f-g)$ is precisely $D_f \cap D_g$, the intersection of the individual domains. This guarantees that both $f(x)$ and $g(x)$ are defined for all $x$ in the domain of the combined function.

Multiplication and Division of Functions: Domain and Explicit Operation

For two functions $f$ and $g$ with domains $D_f$ and $D_g$, the product function $(fg)$ is defined on $D_f \cap D_g$ by

$$(fg)(x) = f(x)\cdot g(x).$$

The quotient function $\left(\dfrac{f}{g}\right)$ is defined at all $x\in D_f\cap D_g$ provided $g(x)\ne 0$. The explicit operation is

$$\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)} \qquad \text{for all } x\in D_f \cap D_g \text{ and } g(x) \ne 0.$$

Hence, the domain of $\left(\dfrac{f}{g}\right)$ is $D_f\cap D_g \setminus \{x:g(x)=0\}$, i.e., the set of common domain points for which the denominator remains nonzero.

Explicit Computation of Domains Under Function Algebra

For each operation above, explicit computation of the domain requires

(i) Determination of $D_f$ and $D_g$ individually, via the internal structure of $f$ and $g$.

(ii) Forming the intersection $D_f \cap D_g$ for addition, subtraction, and multiplication.

(iii) For division, excluding all $x$ for which $g(x)=0$ from $D_f\cap D_g$.

For further background on domain rules for various function types, refer to Types of Functions.

Worked Examples in Algebra of Functions: Fully Detailed Solutions

Example 1: Let $f(x) = x^2 + 2$, $g(x) = x + 1$. Find $(f+g)(x)$ and its domain.

Given: $f(x) = x^2 + 2$ has domain $\mathbb{R}$, $g(x) = x+1$ has domain $\mathbb{R}$.

Substitution: Compute $f(x) + g(x)$:

$\displaystyle (f+g)(x) = f(x) + g(x) = (x^2 + 2) + (x + 1)$

Simplification: Combine like terms:

$x^2 + x + (2+1) = x^2 + x + 3$

Final result: $(f+g)(x) = x^2 + x + 3$ with domain $\mathbb{R}$.

Example 2: If $f(x) = x+2$, $g(x) = x^2-3x+2$, determine $(f/g)(x)$ and its domain.

Given: $f(x) = x+2$ ($D_f = \mathbb{R}$), $g(x) = x^2 - 3x + 2$ ($D_g = \mathbb{R}$).

Substitution:

$\displaystyle (f/g)(x) = \dfrac{x+2}{x^2-3x+2}$

Simplification: Factorise denominator:

$x^2 - 3x + 2 = (x-2)(x-1)$

$\displaystyle (f/g)(x) = \dfrac{x+2}{(x-2)(x-1)}$

Domain determination: Denominator cannot be zero:

Set $(x-2)(x-1)\ne 0 \implies x\ne2$, $x\ne1$

Final result: $(f/g)(x) = \dfrac{x+2}{(x-2)(x-1)}$, domain $D = \mathbb{R}\setminus\{1,2\}$.

Domain Analysis for Composed Function Types

For functions involving radicals, denominators, or compositions, the domain computation follows function-specific criteria:

Radicals: For $f(x)=\sqrt{h(x)}$, the domain is $\{x: h(x) \ge 0\}$ when considering real-valued functions.

Rational Functions: For $f(x) = \dfrac{p(x)}{q(x)}$, the domain is all $x\in \mathbb{R}$ such that $q(x)\ne0$.

For an explicit worked example, suppose $f(x) = \dfrac{1}{x-2}$ and $g(x) = \sqrt{x-1}$.

Step 1: $D_f = \mathbb{R} \setminus \{2\}$

Step 2: $D_g = [1,\infty)$

Sum: The domain is $D_f\cap D_g = [1,2)\cup(2,\infty)$

Quotient: Require $g(x)\ne0\implies x-1>0\implies x>1$, $x\ne2$

Domain: $(1,2)\cup(2,\infty)$

For foundational insights on algebraic structures, see Introduction to Algebra.

Algebraic Principles and Notational Summary

All arithmetic operations on functions are conducted pointwise, i.e., for a real variable $x$, the operation applies separately at each $x$ in the intersection of function domains. For two functions $f$ and $g$ with common domain $D$:

$$(f+g)(x) = f(x) + g(x)$$

$$(f-g)(x) = f(x) - g(x)$$

$$(fg)(x) = f(x)\cdot g(x)$$

$$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad g(x)\ne 0$$

Note that for the quotient, $g(x)$ must remain nonzero throughout the considered domain. For further practice and advanced topics, reference can be made to supplementary chapters such as Properties of Determinants.