Writing Inverse Functions to Solve Problems
5 min
Narrative
In this Warm-up, students make sense of a linear function with a negative rate of change, set in a familiar context. Students have seen similar relationships between the amount of water in a tank and time in an earlier unit (on solving and graphing equations). Here, they think about the quantities in terms of input and output, interpret statements in function notation, and sketch a graph of the function.
The work here prepares students to find and interpret the inverse of functions written in function notation and to use inverse functions to solve problems.
Launch
Arrange students in groups of 2. Provide access to scientific or four-function calculators, if requested.
Student Task
A tank contained some water. The function w represents the relationship between t, time in minutes, and the amount of water in the tank in liters. The equation w(t)=80−2.5t defines this function.
-
Discuss with a partner:
- How is the water in the tank changing? Be as specific as possible.
- What does w(t) represent? Is w(t) the input or the output of this function?
- Sketch a graph of the function. Be sure to label the axes.
Sample Response
- Sample response:
- The tank started with 80 liters of water, but the water is being drained at a constant rate of 2.5 liters a minute. t is time since the draining begins.
- w(t) represents the amount of water left in the tank, in liters, as a function of time in minutes, t. w(t) is the output.
- See graph.
Activity Synthesis (Teacher Notes)
Select students to share their responses, including their graph.
Make sure students recall that in an equation such as w(t)=80−2.5t, w(t) is the output when the input is t. And they can interpret the equation in terms of the situation and see why the graph is a downward-slanting line.
Standards
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- HSF-IF.B.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- HSF-IF.A.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- HSF-IF.B.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <span>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.</span>
10 min
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Writing Inverse Functions to Solve Problems
5 min
Narrative
In this Warm-up, students make sense of a linear function with a negative rate of change, set in a familiar context. Students have seen similar relationships between the amount of water in a tank and time in an earlier unit (on solving and graphing equations). Here, they think about the quantities in terms of input and output, interpret statements in function notation, and sketch a graph of the function.
The work here prepares students to find and interpret the inverse of functions written in function notation and to use inverse functions to solve problems.
Launch
Arrange students in groups of 2. Provide access to scientific or four-function calculators, if requested.
Student Task
A tank contained some water. The function w represents the relationship between t, time in minutes, and the amount of water in the tank in liters. The equation w(t)=80−2.5t defines this function.
-
Discuss with a partner:
- How is the water in the tank changing? Be as specific as possible.
- What does w(t) represent? Is w(t) the input or the output of this function?
- Sketch a graph of the function. Be sure to label the axes.
Sample Response
- Sample response:
- The tank started with 80 liters of water, but the water is being drained at a constant rate of 2.5 liters a minute. t is time since the draining begins.
- w(t) represents the amount of water left in the tank, in liters, as a function of time in minutes, t. w(t) is the output.
- See graph.
Activity Synthesis (Teacher Notes)
Select students to share their responses, including their graph.
Make sure students recall that in an equation such as w(t)=80−2.5t, w(t) is the output when the input is t. And they can interpret the equation in terms of the situation and see why the graph is a downward-slanting line.
Standards
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- HSF-IF.B.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
- HSF-IF.A.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- HSF-IF.B.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <span>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.</span>