Section 3.4 - Piecewise-defined Functions

Section Objectives

1. Define and evaluate piecewise functions.
2. Sketch the graph of a piecewise-defined function.

Piecewise Functions

A piecewise-defined function is a function defined by two or more formulas, where any particular formula is valid only for a unique portion of the function's overall domain.

For example...

When evaluating or graphing piecewise-defined functions, be sure to use only the formula that applies to the portion of the domain that is under consideration.

Examples

• Refer to the function $f$ given above. Evaluate $f(0)$, $f(-5)$, and $f(3)$. (Solution)

• Consider the following function:

  $$g(x)=\left\{ \begin{array}{rc} 1, & \quad x < -1 \\ -x, & \quad -1 < x < 1 \\ -1, & \quad x>1 \end{array} \right.$$
  
  

  Sketch the graph of $g$. Determine the domain and range of $g$. (Solution)

Continuity

A function is said to be continuous if its graph can be drawn without picking up your pencil. Piecewise-defined functions can be discontinuous at the break points even if the individual formulas (or pieces) define continuous functions.

Examples

• Consider the following function:

  $$h(x)=\left\{ \begin{array}{cc} 6, & \quad x < -4 \\ -10+x^2, & \quad -4 \le x < 4 \\ 2x-2, & \quad x\ge 4 \end{array} \right.$$
  
  

  Sketch the graph of $h$. Is $h$ a continuous function? (Solution)

• Sketch the graph of $f$:

  $$f(x)=\left\{ \begin{array}{cc} x-3, & \quad -2 \le x < 3 \\ 9-2x, & \quad x \ge 3 \end{array} \right.$$

(Solution)

For further help...

There are lots of resources available to help you with transformations.

1. Do a Google search for "piecewise functions".
2. Desmos and Geogebra can be used to graph piecewise functions. I'd be happy to help you with the syntax. For the example right above, the Desmos syntax is $y=\{-2<=x<3:x-3,x>=3:9-2x\}$.