piecewise functions objective: students will be able to graph, write and evaluate piecewise...
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Piecewise Functions
• Objective: Students will be able to graph, write and evaluate piecewise functions.
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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
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• 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.
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Piecewise FunctionsA function made up of a combination of
equations, each corresponding to a part of the domain.
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1 ,13
1 ,12
xifx
xifxxf
•One equation gives the value of f(x) when x ≤ 1•And the other when x>1
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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
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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)
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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.
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Graph:
1, 2( )
1, 2
x if xf x
x if x
Point of Discontinuity
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Step Functions
43 ,432 ,321 ,210 ,1
)(
xifxifxifxif
xf
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43,432,321,210,1
)(
xifxifxifxif
xf
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Graph :
01,412,323,234,1
)(
xifxifxifxif
xf
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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
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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
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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?
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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|
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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
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Writing a piecewise function
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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.
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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?
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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?