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.

Definition: Piecewise Function – a function defined by two or more functions over a specified domain.

f(x) = What do they look like? x2 + 1 , x  0 x – 1 , x  0 You can EVALUATE piecewise functions. You can GRAPH piecewise functions.

f(x) = Evaluating Piecewise Functions: Evaluating piecewise functions is just like evaluating functions that you are already familiar with. Let’s calculate f(2). f(x) = x2 + 1 , x  0 x – 1 , x  0 You are being asked to find y when x = 2. Since 2  0, you will only substitute into the second part of the function: f(2) = 2 – 1 = 1

f(x) = Let’s calculate f(-2). x2 + 1 , x  0 x – 1 , x  0 You are being asked to find y when x = -2. Since -2  0, you will only substitute into the first part of the function: f(-2) = (-2)2 + 1 = 5

f(x) = Your turn: 2x + 1, x  0 2x + 2, x  0 Evaluate the following: ? -3 f(5) = 12 ? f(1) = 4 ? f(0) = ? 2

f(x) = One more: 3x + 2, x  -2 -x , -2  x  1 x2 – 7x, x  1 Evaluate the following: f(-2) = ? 2 f(3) = -12 ? f(-4) = -10 ? f(1) = -6 ?

 f(x) = Graphing Piecewise Functions: x2 + 1 , x  0 x – 1 , x  0 Determine the shapes of the graphs. Parabola and Line Determine the boundaries of each graph.       Graph the line where x is greater than or equal to zero. Graph the parabola where x is less than zero. 

  f(x) = Graphing Piecewise Functions: 3x + 2, x  -2 -x, -2  x  1 Determine the shapes of the graphs. Line, Line, Parabola Determine the boundaries of each graph.          

Special Piecewise Function A piecewise function that is constant for each interval of its domain, such as the ticket price function, is called a step function. You can describe piecewise functions with a function rule.

Graph this function: This special “staircase” step function is called the greatest integer function which can be written as f(x)=[x]. In this function all non-integers are rounded down to the nearest integer.

Piecewise functions are easy to graph with your Calculator. Select Dot mode Turn off other functions Y1=(x+3)(-2 < x)(x < 1)+(3)(x = 1)+(-x + 3)(x > 1)