# Lesson 17

Interpreting Function Parts in Situations

These materials, when encountered before Algebra 1, Unit 4, Lesson 17 support success in that lesson.

## 17.1: Math Talk: Function Evaluation (10 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for evaluating inputs for a given output of a function. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to work with inverse functions.

### Student Facing

Mentally find the value of \(x\) for the given function value using the function: \(f(x) = 3(x-2)\)

\(f(x) = 9\)

\(f(x) = 210\)

\(f(x) = 10\)

\(f(x) = 0\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

## 17.2: A Long Car Trip (15 minutes)

### Activity

In this activity students make sense of a linear function in context. They interpret the rate of change and function values given inputs that are in and out of the domain that makes sense for the situation. In the associated Algebra 1 lesson students use inverse functions to discover information about situations. This activity supports students by asking them to think about a situation based on the function that represents it.

### Student Facing

On a long car trip, the distance on the odometer (in miles) is a function of time (in hours after the trip begins) given by the equation \(d(t) = 34t + 45,\!233\).

- What is the rate of change for the function? What does it mean in this situation?
- What is the value of \(d(0)\)? What does it mean in this situation?
- What is the value of \(d(\text{-}1)\)? What does it mean in this situation?
- When is \(d(t) = 45,\!800\)?
- Do each of the values make sense? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to recall the connection between functions and the situations they describe. Select students to share their solutions. Ask students,

- “What is a reasonable domain for the function in this situation? Explain your reasoning.” (\(0 \leq x \leq 10\) because it only makes sense after the trip has started and the car trip probably wouldn’t go longer than 10 hours.)
- “Is a linear function realistic for this situation? Explain your reasoning.” (It is not very realistic because the car will likely slow down or stop due to things like traffic or stop lights or similar things rather than maintain a constant 34 mph for the entire trip.)
- “Even though the linear function may be unrealistic to completely describe the car trip, is there any use to using a linear function to model a car trip? Explain your reasoning.” (There is some use. If the average speed of the car is 34 mph, a linear function like this may give an approximate reading on the odometer which could be useful to know when the car may be running low on gas or need maintenance.)

## 17.3: A Warehouse and Highway (20 minutes)

### Activity

In this activity students use a description of a situation to write a function representing the situation then describe the domain and range of the function and evaluate what input is associated with a given output. In the associated Algebra 1 lesson students use functions and their inverses to answer questions about situations. Students are supported in this lesson by recalling the connection of functions to situations and using some inverse thinking to get a desired input for a given output.

### Student Facing

- A warehouse in a factory initially holds 2,385 items and receives all of the items made in production throughout a day. During a particular day, the factory produces 150 items per hour to put into the warehouse. Write a function, \(f\), to represent the number of items in the warehouse at time \(t\) after production begins for the day.
- What are the units for \(t\)?
- What is the domain of the function? Explain your reasoning.
- What is the range of the function? Explain your reasoning.
- What is the value of \(t\) when \(f(t) = 3,\!000\)? What does that mean in this situation?

- During a focused effort on building new infrastructure for 3 years, a company can build 0.8 miles of highway per day. The company has already built 12 miles of highway before the focused effort. Write a function, \(g\), to represent the length of highway built by the company as a function of \(t\) during the focused effort.
- What are the units for \(g(t)\)?
- What is the domain of the function? Explain your reasoning.
- What is the range of the function? Explain your reasoning.
- What is the value of \(t\) when \(g(t) = 400\)? What does that mean in this situation?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of this discussion is to draw additional connections between a function and the situation it describes. Select students to share their solutions and reasonings. Ask students,

- “How did you think about writing the function to represent the situation?” (First I recognized what was already there when the time started and used that as the constant term. Then I found the amount that was added after each unit of time and used that rate of change as the coefficient of \(x\).)
- “How did you find the value of \(t\) given the function value? For the question here, is it easier to use the inverse to find the value of \(t\) or use the original function?” (To find \(t\), I substituted the value I was given for the function in my function equation and solved for \(t\). Because there was only 1 question, it was easier for me to not use the inverse. If I needed to find the value for \(t\) for several function values, it might make sense to use the inverse like in the warm-up.)