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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 Functions A function made up of a combination of equations, each corresponding to a part of the domain.
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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 First you have to figure out which equation to use You NEVER use both X=0 This one fits Into the top equation So: 0+2=2 f(0)=2 X=2 This one fits here So: 2(2) + 1 = 5 f(2) = 5 X=4 This one fits here So: 2(4) + 1 = 9 f(4) = 9
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Graph: 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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x=1 is the breaking point of the graph. To the left is the top equation. To the right is the bottom equation.
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Graph: Point of Discontinuity
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Step Functions
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Graph :
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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
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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 Absolute An Absolute Value FunctionAn 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 Functions the graphs resemble a set of stair steps The greatest integer function is a step function For 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?
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