Formal Definition of a Function

Examples, videos, and solutions to help Grade 8 students know that a function assigns to each input exactly one output.

New York State Common Core Math Grade 8, Module 5, Lesson 2

Worksheets for Grade 8

Lesson 2 Student Outcomes

• Students know that a function assigns to each input exactly one output.
  
• Students know that some functions can be expressed by a formula or rule, and when an input is used with the formula, the outcome is the output.

Lesson 2 Student Summary

A function is a rule that assigns to each input exactly one output. The phrase “exactly one output” must be part of the definition so that the function can serve its purpose of being predictive.
Functions are sometimes described as an input-output machine. For example, given a function the input is time and the output is the distance traveled in seconds.

Lesson 2 Classwork

Shown below are the table from Example 2 of the last lesson and another table of values. Make a conjecture about the differences between the two tables. What do you notice?

Exercise 1

1. Let y be the distance traveled in time t. Use the function y = 16t² to calculate the distance the stone dropped for the given time t.
   a. Are the distances you calculated equal to the table from Lesson 1?
   b. Does the function y = 16t² accurately represent the distance the stone fell after a given time ? In other words, does the function assign to the correct distance? Explain.

Exercises 2–5
2 - 4. Can the table shown below represent a function? Explain.
5. It takes Josephine 34 minutes to complete her homework assignment of 10 problems. If we assume that she works at a constant rate, we can describe the situation using a function.
a. Predict how many problems Josephine can complete in 25 minutes.
b. Write the two-variable linear equation that represents Josephine’s constant rate of work.
c. Use the equation you wrote in part (b) as the formula for the function to complete the table below. Round your answers to the hundredths place.
After minutes, Josephine was able to complete problems, which means that she was able to complete problem, then get about halfway through the next problem.
d. Compare your prediction from part (a) to the number you found in the table above.
e. Use the formula from part (b) to compute the number of problems completed when x = -7. Does your answer make sense? Explain.
f. For this problem we assumed that Josephine worked at a constant rate. Do you think that is a reasonable assumption for this situation? Explain.

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