Section 2.1What is a function?

Objectives:• To understand functions and function notation.• To review domain.• To understand how to evaluate functions and piecewise functions.• To understand the difference quotient.

Functions

• A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the curve more than once.

• Vertical Line Test – when given a graph, use this to verify if the curve is a function

Ex 1. Use the vertical line test to determine if the following graphs are functions.

FunctionsA function is a relationship between two variables such

that each value of the first variable is paired with exactly one value of the second variable.

Function notation: f(x) = yf(2) means evaluate the given function when

x=2

Ex 2. Given 2

1( )

1f x

x

Evaluate:

a)f(3)

b)f(-2)

c)f( )5

Getting Information from a Graph

• The values of a function are represented by the height of its graph above the x-axis.

– So, we can read off the values of a function from its graph.

Ex. 3 Find the Values of a Function from a Graph

The function T graphed here gives the temperature between noon and 6 P.M. at a certain weather station.

(a) Find T(1), T(3), and T(5).(b) Which is larger, T(2) or T(4)?

Class Work

1)

a) Find f(50).

b) Find f(100).

2.) Evaluate at the given values.

a)f(-2) b) f(4) c) f( ½ )

Class Work2( ) 3 5f x x x

Piecewise Functions

A piecewise function is defined by different formulas on different parts of the domain.

Ex 4. A cell phone plan costs $39 a month.• The plan includes 400 free minutes and charges

20¢ for each additional minute.• The monthly charges are a function of the

number of minutes used, given by

Find C(100), C(400), and C(480)

39 0 400( )

39 0.2( 400) 400

if xC x

x if x

Class Work3. Evaluate the piecewise functions at the indicated

values.a) f(-5)

b) f(3)

c) f(0)

d) f(1)

2

3 0

( ) 1 0 2

2 2

x if x

f x x if x

x if x

Difference Quotient

Ex 4. Evaluate f(x) = 3x – 1 at the following values.

a) f(a)

b) f(a + h)

c) ( ) ( )f a h f a

h

Class Work

4. Find the difference quotient, for the function:

( ) ( )f a h f a

h

2( ) 2 9f x x

5. Use the function to evaluate the indicated expression and simplify.

f(x) = 3x – 1 a) f(2x) =

b) 2f(x) =

c) f(x2) =

d) (f(x))2=

Pg.155 (13-27) odd, 29, 31, 35, 59, 60, 67, 68Pg. 167 (23-26, 55-60) all