Piecewise Functions

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.

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 First you have to figure out which equation to use You NEVER use both X=0 This one fits into the top equation X=2 X=4 This one fits here This one fits here So: 2(2) + 1 = 5 f(2) = 5 0+2=2 f(0)=2 2(4) + 1 = 9 f(4) = 9

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)

x=1 is the breaking point of the graph. To the left is the top equation. To the right is the bottom equation.

Graph: Point of Discontinuity

You have a summer job that pays time and a half for overtime You have a summer job that pays time and a half for overtime. That is, 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 $7.00. Write and a graph a piecewise function that gives you 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?

GARAGE RATES $3 per half hour $8 maximum for 12 hours Write a piecewise function for the parking garages shown on the sign. What is the domain and range of the function?

Step Functions

Graph :

Special Step Functions Two particular kinds of step functions are called ceiling functions ( f (x)= and floor functions ( f (x)= ). In a ceiling function, all nonintegers are rounded up to the nearest integer. An example of a ceiling function is when a phone service company charges by the number of minutes used and always rounds up to the nearest integer of minutes.

Special Step Functions In a floor function, all nonintegers are rounded down to the nearest integer. The way we usually count our age is an example of a floor function since we round our age down to the nearest year and do not add a year to our age until we have passed our birthday. The floor function is the same thing as the greatest integer function which can be written as f (x)=[x].