Lesson 12

Piecewise Functions

Let’s look at functions that are defined in pieces.

Problem 1

A parking garage charges $5 for the first hour, $10 for up to two hours, and $12 for the entire day. Let \(G\) be the dollar cost of parking for \(t\) hours.

  1. Complete the table.
  2. Sketch a graph of \(G\) for \(0 \leq t \leq 12\).
    Blank coordinate grid, origin O. Horizontal axis, time, hours, from 0 to 12 by 2’s. Vertical axis, cost, dollars, from 0 to 14, by 2’s.
  3. Is \(G\) a function of \(t\)? Explain your reasoning.
  4. Is \(t\) a function of \(G\)? Explain your reasoning.
\(t\) (hours) \(G\) (dollars)
0  
\(\frac 12\)  
1  
\(1\frac 3 4\)  
2  
5  

Problem 2

Is this a graph of a function? Explain your reasoning.

Piece wise function.

Problem 3

Use the graph of function \(g\) to answer these questions.

  1. What are the values of \(g(1)\), \(g(\text-12)\), and \(g(15)\)?
  2. For what \(x\)-values is \(g(x)=\text{-} 6\)?
  3. Complete the rule for \(g(x)\) so that the graph represents it.

    \(\displaystyle g(x) =\ \begin{cases} \text{-}10, & \text{-}15\leq x< \text{-}10 \\ \underline{\hspace {8mm}}, & \text{-}10\leq x<\text{-}8 \\ \text{-}6, & \underline{\hspace {8mm}}\leq x<\text{-}1 \\ \underline{\hspace {8mm}}, & \text{-}1\leq x<1 \\ 4, & \underline{\hspace {8mm}}\leq x<\underline{\hspace {8mm}} \\ 8, & 10\leq x<15 \\ \end{cases} \)

    piecewise function. 

Problem 4

This graph represents Andre’s distance from his bicycle as he walks in a park. 

Graph of Andre’s distance from his bicycle, coordinate plane, origin \(O\). 
  1. For which intervals of time is the value of the function decreasing?
  2. For which intervals is it increasing?
  3. Describe what Andre is doing during the time when the value of the function is increasing. 
(From Unit 4, Lesson 6.)

Problem 5

The temperature was recorded at several times during the day. Function \(T\) gives the temperature in degrees Fahrenheit,  \(n\) hours since midnight.

Here is a graph for this function.

  1. Describe the overall trend of temperature throughout the day.

    Discrete graph of temperature over time, coordinate plane, origin O.
  2. Based on the graph, did the temperature change more quickly between 10:00 a.m. and noon, or between 8:00 p.m. and 10:00 p.m.? Explain how you know.
(From Unit 4, Lesson 7.)

Problem 6

Explain why this graph does not represent a function.

A graph, origin O. A curve, shaped like a U, opening to the right.
(From Unit 4, Lesson 8.)