Section 1.4 Functions

Chapter 1: Graphs and Functions

Section 1.4: Functions

Objectives:

• Use functional notation and find function values.

• Find the domain and range of a function.

• Identify the graph of a function and extract information from it.

• Solve applied problems using functions.

Definition of a Function

• A function is a special relationship where each element x in a set corresponds to a unique element y in another set.

• Independent Variable: x

• Dependent Variable: y

Function Properties

• A function assigns exactly one element of Y to each element of X.

• Domain: Set X (inputs of the function).

• Range: Set Y corresponding to elements of X.

Function Notation

• Denoted as f(x) or "f of x."

• It represents the value of the function at x.

Relations and Functions

• Relations can be expressed in ordered pairs.

• A function is defined when every x-coordinate (domain) matches with exactly one y-coordinate (range).

Examples of Determining Functions

• Identify whether relations given as ordered pairs or equations define a function.

• Use correspondence diagrams to visualize.

• Relation is not a function if two pairs have the same x-value but different y-values.

Graphical Representation of Functions

• Both tables and graphs can describe functions.

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

Evaluating Functions and Domain/Range

• To find the domain, look for valid x-values; the range is found by determining valid y-values from the graph.

• Finding function values: f(x) can be evaluated as follows:

  • Example: If f(x) = 2x + 3, then f(1) = 2(1) + 3 = 5.

Applied Examples

• Practical examples, such as evaluating equations or determining areas, reinforce understanding of functions and their properties.

  • Cholesterol-Reducing Drugs: Illustrates max concentration with functions. Example: C(x) = ax^2 + bx + c (parabolic equation for concentration over time).

  • Cost of Fiber Cable: Shows how to formulate cost functions for decision-making using linear models.Example: If the cost C(x) is represented by C(x) = mx + b (where m is the cost per unit and b is a fixed cost).

Economic Functions

• Functions used in economic contexts typically involve cost, demand, and profit analysis.

• Profit Function: P(x) = R(x) - C(x) (where R is revenue and C is cost). Writing and evaluating these functions are crucial to understanding business operations.