Functions

Exploring the Idea of a Function

In mathematics and the physical sciences, a function describes how one quantity depends on another. Functions are fundamental for modeling relationships between variables in real-world and abstract contexts.

• Examples in daily life:

  • Height as a function of age

  • Temperature as a function of date

  • Postage cost as a function of package weight

• Scientific examples:

  • Area of a circle as a function of its radius

  • Number of bacteria in a culture as a function of time

  • Weight of an astronaut as a function of elevation

Definition of a Function (Verbal and Mathematical)

A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B.

• Independent variable: The input value, usually denoted by x

• Dependent variable: The output value, usually denoted by f(x)

The notation f(x) is read as "f of x" or "f at x" and represents the image of x under f.

• Domain: The set containing all possible input values (x elements)

• Range: The set of all possible output values, that is, B = ext{range of } f = ig\{ f(x) ig| x \\in A ig\\}

Function as a Machine

A function can be visualized as a machine that takes an input, processes it according to a rule, and produces an output.

• Input: The value fed into the function

• Output: The result after applying the function's rule

Function Notation

The equation y = f(x) is called function notation. Here, f is the name of the function, x is the input, and y is the corresponding output.

• Example: If f(x) = x^2, then for x = 3, f(3) = 9

Arrow Diagram Representation

An arrow diagram is a way to picture a function by associating each input from set A to its corresponding output in set B.

• Function property: Each input is associated with exactly one output.

• Non-function: If an input is associated with more than one output, the relation is not a function.

Different Ways to Represent a Function

Functions can be represented in several ways, each providing unique insights:

• Verbally: By a description in words (e.g., "double the input and add three")

• Algebraically: By an explicit formula (e.g., f(x) = 2x + 3)

• Graphically: By a graph (plotting (x, f(x)) pairs)

• Numerically: By a table of values (listing input-output pairs)

Often, it is useful to switch between representations to better understand the function's behavior. Some functions are more naturally described by one method than another.

Summary Table: Ways to Represent Functions

Key Properties of Functions

• Each input has exactly one output.

• Domain: All possible input values

• Range: All possible output values

• Function notation:

Example: Area of a Circle

• Verbal: The area depends on the radius.

• Algebraic:

• Graphical: Plot vs.

• Numerical: Table of and values

Additional info:

• These notes are suitable for an introductory Precalculus course, focusing on the concept and representation of functions.