Piecewise Functions

• Objective: Students will be able to graph, write and evaluate piecewise functions.

Relation – a mapping of input and output values

Function - a relation that has a unique output for each input ( every x has a unique y )

vertical line test – a relation is a function if and only if no vertical line intersects the graph of the relation at more than one point.

Domain – the input values, x values, independent variable

Range – the output values, y values, dependent variable

• Up to now, we’ve been looking at functions represented by a single equation.

• In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain.

• These are called piecewise functions.

Piecewise FunctionsA function made up of a combination of

equations, each corresponding to a part of the domain.

1 ,13

1 ,12

xifx

xifxxf

•One equation gives the value of f(x) when x ≤ 1•And the other when x>1

Evaluate f(x) when x=0, x=2, x=4

2 ,12

2 ,2)(

xifx

xifxxf

•First you have to figure out which equation to use•You NEVER use both

X=0This one fitsInto the top equation

So:0+2=2f(0)=2

X=2This one fits hereSo:2(2) + 1 = 5f(2) = 5

X=4

This one fits hereSo:2(4) + 1 = 9f(4) = 9

Graph:

1 ,3

1 ,)( 2

321

xifx

xifxxf

•For all x’s < 1, use the top graph (to the left of 1)

•For all x’s ≥ 1, use the bottom graph (to the •right of 1)

12

3, 1

2( )3, 1

x if xf x

x if x

x=1 is the breakingpoint of the graph.

To the left is the topequation.

To the right is thebottom equation.

Graph:

1, 2( )

1, 2

x if xf x

x if x

Point of Discontinuity

Step Functions

43 ,432 ,321 ,210 ,1

)(

xifxifxifxif

xf

43,432,321,210,1

)(

xifxifxifxif

xf

Graph :

01,412,323,234,1

)(

xifxifxifxif

xf

Graphing a Piecewise Function

Graph each part of the function individually but put them on the same graph

Graph this function:

-x + 3, if x ≥ 1

1,2

3

2

1)( ifxxxf

Evaluating a Piecewise Function Evaluate f(x) when:

x = 0, 2 and 4

f(x) = x + 2, if x <2 2x + 1, if x ≥ 2

Snowstorm

• During a ten hour snowstorm it snows at a rate of 1 inch per hour for the first 3 hours, at a rate of 2 inches per hour for the next six hours and 1 inch per hour for the final hour.

• Write and graph a piecewise function that gives the depth of snow during the snowstorm. How many inches of snow accumulate from the storm?

The Absolute Value Function

The Absolute

An Absolute Value Function is a famous Piecewise Function.It has two pieces:•below zero: -x

•from 0 onwards: x

f(x) = |x|

Step Functionsthe graphs resemble a set of stair steps

The greatest integer function is a step functionFor every real number x, g(x) is the greatest integer less

than or equal to x

Writing a piecewise function

Using a step function

A parking garage charges $3 for the first hour and $8 for a maximum of twelve hours

($3 for the first hour and $8 for hours 2-12)Write and graph a piecewise function for the parking charges.

You have a summer job that pays time and a half for overtime. If you work more than 40 hours per week, your hourly wage for the extra hours is 1.5 times your normal hourly wage of $7.

Write and graph a piecewise function that gives your weekly pay P in terms of the number of hours, h, you work.

How much will you get paid if you work 45 hours?

14.

You have a summer job that pays time and a half for overtime. If you work more than 40 hours per week, your hourly wage for the extra hours is 1 1/2 times your normal hourly wage of $10.

a.) Write a piecewise function that gives your weekly pay P in terms of the number of hours, h, you work.

b.) How much will you get paid if you work 46 hours?