Answer the following questions and if required, do calculations to please show your work:
1. What is a function?
2. What is a linear function?
3. What form does a linear function take? (e.e., What is the standard mathematical notation of a linear function?)
4. What is the formula for determining the slope of a line?
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 the number of cups, and "P" for the price you charge. Assume the function is linear.
6. Take a look at the table below and compute an equation for f(x).
x -3 -2 1 3 4
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F(x) 0 3 12 18 21
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 determining if they are functions.
a. f(x) = x^8
b. f(x) = 45 if x>2 otherwise f(x) = -4
c. f(x) = 4 if x>0 or f(x) = -4 if x<0 or f(x) = 4 or -4 if x = 0
8. 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.
b. The probability of getting into a car accident as a function of the speed at which you drive.
c. A person's height as a function of the person's age (from age 0 to 100).