Functions: Definitions, Representations, and Operations in Calculus

Functions and Their Properties

Definition of a Function

A function is a fundamental concept in calculus, describing a relationship between two sets where each input from the first set is assigned exactly one output in the second set.

• Formal Definition: A function f from set X to set Y is denoted as f: X → Y. For every x ∈ X, there exists a unique y ∈ Y such that f(x) = y.

• Notation: $f: X \to Y$

• Example: If X = {1, 2, 3} and Y = {a, b, c}, a function could be f(1) = a, f(2) = b, f(3) = c.

One-to-One (Injective) Functions

A function is one-to-one (injective) if different inputs always produce different outputs.

• Definition: f is injective if for any x1, x2 ∈ X, f(x1) = f(x2) implies x1 = x2.

• Example: f(x) = x + 1 is injective on the set of real numbers.

• Non-example: f(x) = x^2 is not injective on R because f(2) = f(-2) = 4.

Representing Functions

Functions can be represented in several ways to facilitate understanding and analysis.

• Table: Listing input-output pairs.

• Graph: Plotting y = f(x) on a coordinate plane.

• Formula: Expressing the relationship algebraically, e.g., f(x) = x^2.

• Example: The graph of f(x) = x^2 is a parabola.

Domain and Range of Functions

Domain

The domain of a function is the set of all possible input values for which the function is defined.

• Notation: $\text{dom}(f)$

• Example: For f(x) = x^2, dom(f) = \mathbb{R}.

Range

The range of a function is the set of all possible output values.

• Notation: $\text{range}(f)$

• Example: For f(x) = x^2, range(f) = [0, \infty).

Finding Domain and Range

To determine the domain and range, analyze the formula and consider restrictions such as division by zero or square roots of negative numbers.

• Example: For f(x) = \sqrt{x}, dom(f) = [0, \infty), range(f) = [0, \infty).

• Quadratic Example: For f(x) = (x-3)^2 + 1, dom(f) = \mathbb{R}, range(f) = [1, \infty).

Combining Functions

Operations on Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Addition: $(f+g)(x) = f(x) + g(x)$

• Subtraction: $(f-g)(x) = f(x) - g(x)$

• Multiplication: $(fg)(x) = f(x)g(x)$

• Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

Domain of Combined Functions

The domain of a combined function depends on the domains of the individual functions and the operation used.

• Addition/Subtraction/Multiplication: $\text{dom}(f \pm g) = \text{dom}(f) \cap \text{dom}(g)$

• Division: $\text{dom}(\frac{f}{g}) = \text{dom}(f) \cap \text{dom}(g) \setminus \{x : g(x) = 0\}$

• Composition: $\text{dom}(f \circ g) = \{x \in \text{dom}(g) : g(x) \in \text{dom}(f)\}$

• Example: If $f(x) = x^2$ and $g(x) = \sqrt{x}$, then $\text{dom}(f \circ g) = [0, \infty)$.

Symmetry of Functions

Even and Odd Functions

Functions can exhibit symmetry about the y-axis (even) or the origin (odd).

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain.

• Example (Even): $f(x) = x^2$

• Example (Odd): $f(x) = x^3$

Summary Table: Function Operations and Domains

Operation	Formula	Domain
Addition	$(f+g)(x) = f(x) + g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
Subtraction	$(f-g)(x) = f(x) - g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
Multiplication	$(fg)(x) = f(x)g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
Division	$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$	$\text{dom}(f) \cap \text{dom}(g) \setminus \{x : g(x) = 0\}$
Composition	$(f \circ g)(x) = f(g(x))$	$\{x \in \text{dom}(g) : g(x) \in \text{dom}(f)\}$

Worked Examples

Example 1: Quadratic Function

• Function: $f(x) = (x-3)^2 + 1$

• Domain: $\mathbb{R}$

• Range: $[1, \infty)$

• Graph: Parabola opening upwards, vertex at $(3, 1)$

Example 2: Square Root Function

• Function: $f(x) = \sqrt{x}$

• Domain: $[0, \infty)$

• Range: $[0, \infty)$

Example 3: Composition

• Given: $f(x) = x^2$, $g(x) = \sqrt{x}$

• Composition: $(f \circ g)(x) = f(g(x)) = (\sqrt{x})^2 = x$

• Domain: $[0, \infty)$

Example 4: Division of Functions

• Given: $f(x) = x^2$, $g(x) = x-1$

• Division: $\frac{f}{g}(x) = \frac{x^2}{x-1}$

• Domain: $\mathbb{R} \setminus \{1\}$

Example 5: Symmetry

• Even Function: $f(x) = x^2$

• Odd Function: $f(x) = x^3$

Additional info: These notes cover foundational concepts in calculus related to functions, including definitions, representations, domain and range, operations, and symmetry. The examples and table provide practical context for understanding how functions are manipulated and analyzed in calculus.