Functions: Definitions, Properties, and Graphical Analysis

Chapter 2: Functions

A. Functions

Functions are foundational objects in mathematics that describe a relationship between two sets, typically called the domain and the range. Understanding functions is essential for analyzing mathematical models and solving real-world problems.

Definition of Relation

• Relation: A relation is a correspondence between two sets: a set X (the domain) and a set Y (the range). Each element from the domain corresponds to at least one element from the range.

Definition of Function

• Function: A function from X into Y is a relation that associates each element of X with exactly one element of Y.

Implicit and Explicit Forms of Functions

• Implicit Form: A function is given implicitly if it is defined by an equation in x and y (e.g., 3x + y = 5).

• Explicit Form: If it is possible to solve for y in terms of x, the function is given explicitly (e.g., y = f(x) = -3x + 5).

Finding the Domain of a Function Defined by an Equation

• Start with the domain as the set of all real numbers.

• If the equation has a denominator, exclude any numbers for which the denominator is zero.

• If the equation has a radical with an even index, exclude any numbers for which the expression inside the radical is negative.

Difference Quotient

• Difference Quotient: For a function f at x, the difference quotient is given by: $\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$

Example: Domain and Graphical Questions

• Given f(x) = \frac{x+1}{x+2}:

  • (a) Find the domain of f.

  • (b) Is the point (1, \frac{1}{2}) on the graph of f?

  • (c) If x = 2, what is f(x)? What point is on the graph of f?

  • (d) If f(x) = 2, what is x? What point is on the graph of f?

  • (e) What are the x-intercepts of the graph of f (if any)? What corresponding point(s) are on the graph of f?

Obtaining Information from or about the Graph of a Function

• If (x, y) is a point on the graph of a function f, then y is the value of f at x: y = f(x).

• If y = f(x), then (x, y) is a point on the graph of f.

Example: Graphical Analysis

• Given a graph, answer questions such as:

  • What are f(0), f(\frac{3\pi}{2}), and f(3\pi)?

  • What is the domain and range of f?

  • List the intercepts, intervals of increase/decrease, and values where f(x) is positive or negative.

B. Graph of a Function

Graphs visually represent the relationship between the domain and range of a function. Key features include intercepts, intervals of increase/decrease, and symmetry.

C. Properties of Functions

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

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

• Increasing/Decreasing/Constant: Definitions as given above.

D. Local Maximum/Minimum

• A function f has a local maximum at c if $f(c) \geq f(x)$ for all $x$ near $c$.

• A function f has a local minimum at c if $f(c) \leq f(x)$ for all $x$ near $c$.

E. Extreme Value Theorem

• If a function f is continuous on a closed interval $[a, b]$, then f has an absolute maximum and minimum on $[a, b]$.

F. Example Graphs