Functions: Basics and Applications

Math Lab Instructor Notes

  • Course Level/Topic: Beginning Algebra/Functions
  • Purpose: To reinforce the concept of function and the use of functional notationrather than to teach the concepts.
  • Learning Goals:

Definition of function.

Given either the domain or range value of a function, determine the other using a graph.

Using function notation, evaluate the value for a variety of functions given both numeric and algebraic expressions.

Solve real world application problems involving functions.

Determine the appropriate domain for a real world functional situation.

  • Pre-requisite knowledge:

Evaluating expressions

Reading graphs

Introduction to function, domain and range, and functional notation

Factoring

  • Materials: None
  • Time Required: about 45 minutes if the activity is completed in class
  • Description: This lab looks at input/output pictures to emphasize that a function has only one output for every input even through the output need not be unique. Using functional notation, students determine both range and domain values from a graph and then do the same for a variety of given functional equations. The real world applications include a piecewise function (cell phone costs) and require the student to find a variety of values as well as determining realistic domains.
  • Implementation:

This lab is a good summative activity to reinforce the concept of function and the use of functional notation.

This lab can be used as an in-class follow-up activity after students have been introduced to the concepts.

This activity makes a good take-home quiz.

It students have not seen rational functions,question #7 can be eliminated.

  • Follow Up/ Discussion Questions:

Have the students calculate their own cell phone rates against that of a competitor.

Discuss why mathematicians prefer functions to relations.

A function is a relation with an important property. For example:

The following are functions

1)Mapping

inputoutput

domainrange

bear

animal

cat

9number

The following are NOT functions

inputoutput

domainrange

bear

animal

cat

number 9

2) Graphs

3)Ordered Pair

(1, -1), (2, -2), (3, -2)(1, -2), (1, -3), (2, 5)

4)Equations

y = x^2x = y^2

1. What property ensures that the relations on the left are functions and those on the right are not?

For every input (domain) value for the left side relations there is only one output (range) value.

2. Define a function.

A function is a relationship where every value of the input variable (domain) has only one output value (range).

  1. The graph to the right shows a function. Determine the following:
  2. 1
  3. 1
  4. 4
  5. 2

In each of the following cases, find the x value that will make the equation true:

  1. If then x = ___4__
  1. If then x = __5_ or x = __3___
  1. If then x = ___2___ or x = ___6___
  1. Consider the following functions…

Evaluate the following:

  1. = -2
  1. An x value for which

x = 4

  1. = ¾
  1. = -64a2 + 120a
  1. Value(s) of n for which g(n)=10

n = -2 or n = 5

  1. = 56
  1. 3k(x) = 6x-9

  1. = h2 + h - 2
  1. Solve x = 0
  1. =
  1. = 0
  1. An d value for which

d = -15

  1. 5g(n) = 5n2 – 15n
  1. What value of d makes v(d) undefined? d = - 5

Function Applications:

5. The humerus is the long bone that makes up the upper part of your arm. The function predicts a person’s height (h) in centimeters based on the length of their humerus (x) in centimeters. About how tall would a person be if their humerus were 32 cm? About how long would a person’s humerus be if that person were 168 centimeters tall? Round to the nearest tenth.

h(32) = 159.48 cm 159.5 cmh = 168 => x 35.1 cm

6. A cell phone company charges their customers based on the number of minutes used….

  • For using 0 to 500 minutes there is only a $45 service charge.
  • For using more than 500 to 1000 minutes there is a $20 service charge plus $0.05 per minute.
  • For using over 1000 minutes there is a $30 service charge plus $0.04 per minute.

Fill in the piecewise function to the right that gives the total charge as a function of the number of minutes used.

Find the following and interpret the result…

C(160) = $45C(2400) = $126C(800) = $60

7. In each case give the domain of the function.

a)All real numbers except 3

b)All real numbers except -1, 0, and 3/2

8. Give the domain of the function that would make sense for real world application.

a) where C is the cost of renting a car and driving it m miles.

b) where h is the height of an object at time t. [0, 5]