3.1 Functions and Function Notation

Chapter 3.1 Functions and Function Notation

Pindling
College Algebra
by Example Series

Key Concept: Understand the fundamental properties of function

Skills to Learn

1. Know how to recognize a function by its equation

2. Know the language use to represent a function

3. Know how to evaluate the Difference Quotient

4. Know how to recognize a function by is graph

5. Know the domain and range of some common functions

Function identification from equation or formula

A function is a relationship that have only one value for y for each values of x

Example: Is a function?

Is a Function: 1 value of y for x: y = 2, x =2
Example: Is a function?

Is a Function: 1 value of y for x:

Example: Is a function?

Not a Function: Many values of x for y:

Example: Is a function?

Not a Function: Many values of x for y:

Example: Is a function?

Is a Function: 1 value of y for x: y = 6, x =2
Example: Is a function?

Is a Function: 1 value of y for x: y = -2, x =2

Example: Is a function?

Is a Function: A horizontal line
Example: Is a function?

Not a Function: A vertical line

Functional Notation

To show that y is a function of x write:

for example

When x = 2,
To show that y is a function of x write:

for example

for example

Example: Given

Evaluate g(x) for x = 2

So g(2) = 15
Example: Given

Evaluate g(x) for x = a+2

So g(a + 2) = 2a + 9

Example: Given

Evaluate g(x) for x = t + 1

So g(t + 1) = t² + 2t + 4

Example: Given

Evaluate g(x) for x = 0

So g(0) = 1

The Difference Quotient

Given the function f(x) and exactly, then the difference quotient is defined as:

Example: Evaluate the difference quotient for

Example: Evaluate the difference quotient for

Vertical Line Test

If a vertical line imposed on a graph intersects the graph only at one point for every value of x, then that graph is a function

A Function

A Graph (not a function)

A Graph (not a function)

A Function

Domain and Range

Domain: Possible values of x of a function

Range: Possible values of y of a function

Example: Find the domain and range of .
Strategy: Cannot have the square root of a negative number: for example.

So

Solve: for x to find the domain

So domain is all values of

To Find the range use to find y

(e.g.

Graphic Solution

Example: Find the domain and range of .
Strategy: All values of x is possible since no value of x gives an impossible solution of y

So domain is x equal all values of x or

Range: the |x| only gives positive numbers, the smallest value of y occurs when x = 0, since y = 0 + 3 = 3 and the highest values of y results when x is either a large negative or positive number.

So range is

Graphic Solution

Example: Find the domain and range of .
Strategy: The denominator cannot be equal to 0; so find values of x that gives denominator = 0 and exclude them from the domain

So , so x = 3 and -3 when denominator = 0

So domain is or or

To find the range we solve the equation of x in terms of y or x=f(y)

or

(not readily seen, but difficult to solve for y)

Graphic Solution ( Range is all values of y)

Example: Find the domain and range of .
No is no value of x that gives an impossible value for y

So domain is x a all values of x.

To find the range we solve for x in terms of y

Cannot have the square root of a negative number so y > = -9 since

So range is