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# Module 2: Functions, Case Study Example

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*Answer the following questions and if required to do calculations please show your work:*

1. What is a function?

A function is a relation of x to y, such that y is a function of x, where for any value of x, there can only be one value of y related to it.

2. What is a linear function?

A linear function of x to y, is a function such that y=ax+b, where a and b are real numbers. A linear function is a function that is represented graphically by a straight line.

3. What form does a linear function take? (I.e., What is the standard mathematical notation of a linear function?)

The standard form of a linear function is y=ax+b where a and b are real number constants.

4. What is the formula for determining the slope of a line?

The general formula for determining the slope of a line is rise/run

Given two points (x1,y1) and (x2,y2) on a line, the formula for the slope is as follows

**slope = (y2-y1)/(x2-x1)**

5. Suppose you have a lemonade stand, and when you charge $1 per cup of lemonade you sell 60 cups. But when you raise your price to $1.50 you only sell 30 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote “C” for number of cups, and “P” for the price you charge. Assume the function is linear.

C(P) = (30-60)/(1.5-1)(P-1) + 60

C(P) = (-30)/0.5(P-1)+60

C(P) = -60P+60+60

**C(P) = -60P + 120**

6. Take a look at the table below and compute an equation for f(x).

X |
-3 | -2 | 1 | 3 | 4 |

f(x) |
0 | 3 | 12 | 18 | 21 |

First, it must be determined if the function is linear. To do this, the slope of every pair of points was computed. It was found that each pair of points had a slope of 3 (work was done using a calculator). Therefore, the function is linear.

Since it is known that the function is linear and has a slope of 3, f(x) is computed by choosing any point. Point (-3,0) is selected.

f(x) = 3[x-(-3)]

f(x) = 3(x+3)

**f(x) = 3x + 9**

7. Which of the following are functions? Explain your reasoning for a, b, and c. Keep the definition of a function strongly in mind as you do this problem, it is not nearly as difficult as it may look. Think about whether the relationship between f(x) and x is consistent with being a function or not and explain your reasoning. Problems b & c are multi part relations so consider all parts when deteriming if they are funcitons.

a. f(x) = x 8

This is not a function because there is no operation between x and 8. The operation can be multiplication, although that is not the conventional way to writer the operation (8 should precede x). If there is an operation of multiplication between the two, then this will become a function.

b. f(x) = 45 if x>2 otherwise f(x) = -4

This is a function since only one value of f(x) is assigned for every value of x. If x is greater than 2, then y is always 45, and if x is 2 or less, f(x) is always -4. The graph of this function is two horizontal lines.

c. f(x) = 4 if x>0 or f(x) = -4 if x<0 or f(x) = 4 or -4 if x = 0

This is not a function because when x is 0, f(x) has 2 values instead of just 1.

48 For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning thoroughly.

a. The __time__ it takes you to get to work as a function the __speed__ at which you drive

This seems like a linear function because the faster you go, the less time it will take you to reach your destination. A linear functional relationship is possible so long as a person’s acceleration is constant.

b. The __probability__ of getting into a car accident as a function of the __speed__ at which you drive

This does not seem like a linear function. Although it’s true that going very fast does make it more likely for you to get in an accident, a certain range of speeds can be equally safe. For example, I would not say that a person driving at 20 mph is already more likely to get into an accident as someone driving at 10mph. Also, many things affect the probability of getting into a car accident. Competent drivers can drive at high speeds safely, while incompetent drivers can drive at lower speeds and still get into accidents.

c. A person’s __height__ as a function of the person’s __age__ (from age 0 to 100)

This is also not a linear function. As a person ages from a baby to adulthood, height may increase linearly. However, the rate will change over time, and then after a certain point, the person’s height will stop increasing although he still gets older.

**Assingment Expectations**

- Differentiate between a function and a linear function.
- Know the mathematical equation of a standard linear function.
- Know how to compute the slope of a line.

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