2.5 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.

Piecewise Functions Piecewise functions are functions that can be represented by more than one equation, with each equation corresponding to a different part of the domain. Piecewise functions do not always have to be line segments. The “pieces” could be pieces of any type of graph. This type of function is often used to represent real-life problems like ticket prices.

Example x + 1, if x < 1 2, if 1 ≤ x ≤ 3 (x-3)2 + 2, if x > 3

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=4 X=2 X=0 This one fits Into the top equation So: 0+2=2 f(0)=2 So: 2(4) + 1 = 9 f(4) = 9 This one fits here So: 2(2) + 1 = 5 f(2) = 5 This one fits here

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

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].