Section 1.3: Functions and Their Representations

Introduction to Functions

Functions are foundational concepts in algebra that describe relationships between two sets, typically called the domain and range. Understanding how to represent, evaluate, and interpret functions is essential for further study in mathematics.

Function Notation

Function notation provides a concise way to express the output of a function for a given input. If f is a function and x is an input, then f(x) denotes the output corresponding to x.

• Key Point: f(x) is read as "f of x" and represents the value of the function at x.

• Example: If f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.

Four Ways to Represent a Function

Functions can be represented in multiple ways to suit different contexts and applications:

• Verbally: Describing the relationship in words.

• Numerically: Using tables of values.

• Graphically: Plotting points or curves on a coordinate plane.

• Algebraically: Writing an explicit formula or equation.

Additional info: The image above shows a blank table (for numerical representation) and a blank grid (for graphical representation), which are commonly used to organize and visualize function data.

Formal Definition of a Function

A function is a relation in which every input is paired with exactly one output. This means that for each element in the domain, there is a unique corresponding element in the range.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (f(x)-values).

Identifying Functions

To determine if a relation is a function, check that each input has only one output. This can be done using tables, graphs, or equations.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Tables: Each input value should appear only once with a single output.

Example: Linear Function (Noah's Job)

Noah earns $10 per hour and can work up to 35 hours per week.

• Formula: , where

• Table of Values: (see image above for blank table to fill in values such as h = 0, 5, 10, ..., 35)

• Graph: Plotting h (hours) on the x-axis and P (pay) on the y-axis yields a straight line.

• Interpretation: This is a function because each number of hours worked yields exactly one pay amount.

• Non-Function Example: If an input (hours) could yield two different pay amounts, it would not be a function.

Set-Builder and Interval Notation

These notations are used to describe the domain and range of functions:

• Set-Builder Notation:

• Interval Notation:

Function Notation and Evaluation

Given a function, you can evaluate it for specific values:

• Example: (Noah earns $300 for 30 hours of work)

• Solving for Input: If , then (but since , this is not possible in context)

Tables and Ordered Pairs

Functions can be represented as tables or sets of ordered pairs:

• Example: If

• Table:

  x	f(x)
  -1	4
  0	3
  1	4
  2	2

• Ordered Pairs:

• Domain:

• Range:

Evaluating Functions

To evaluate a function, substitute the input value into the formula:

• Example: Let

• f(4):

• f(3):

• f(-3): (undefined, since division by zero is not allowed)

• Domain: All real numbers except (since the denominator cannot be zero)