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Function Definitions and Linear Equations

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1. What is a function?

2. What is a linear function?

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

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

5. Which of the following are functions? The last two problems, i.e., b & c, are multi part relations consider all parts when determining whether or not these relations are functions. Explain your reasoning for a, b, and c.

a. f(x) = x + 3

b. f(x) = 73 if x>2 otherwise f(x) = -1

c. f(x) = 79if x>0 or f(x) = -9 if x<0 or f(x) = 9 or -9 if x = 0

6. Suppose you have a lemonade stand, and when you charge $2 per cup of lemonade you sell 100 cups. But when you raise your price to $4 you only sell 50 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. Show your work.

7. Take a look at the table below and write out an equation for f(x). Show your work.

x -3 -2 1 3 4
f(x) 0 3 12 18 21

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. A person's height as a function of the person's age (from age 0 to 100)

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

c. The time it takes you to get to work as a function the speed at which you drive

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Solution Summary

Basic definitions of functions and linear equations. Mathematical models of functions are shown to simulate different situations. Determine whether a given situation can be modeled by a linear function.

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1. What is a function?
A function is a correspondence from one set to another set. The first set is called the domain and the second set is called the range. In a function, if you pick any element in the domain, there is only one element in the range that corresponds to it.

2. What is a linear function?
A function whose graph is a nonvertical straight line.

3. What form does a linear function take? (I.e., What is the standard mathematical notation of a linear function?)
f(x) = mx + b where m is the slope of the line and b is the point where the line crosses the y-axis.

4. What is the formula for determining the slope of a line?
m = (y2 - y1) / (x2 - x1) where m is the slope and (x1, y1) and (x2, y2) are any two different points that are on the line.

5. Which of the following are functions? The last two problems, i.e., b & c, are multi part relations consider all parts when determining whether or not these relations are functions. ...

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