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Algebra equations

1. Determine whether each of the following is a function or not.
(a) f(x) = 1 if x>1
= 0 otherwise
(b) f(x) = 2 if x>0
= -2 if x<0
= 2 or -2 if x = 0
= 0 otherwise
(c) f(x) = 5/x

2. 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 $2 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.

3. Take a look at the table below and write out an equation for f(x)

x -2 -1 0 1 2
f(x) -1 1 3 5 7

4. 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.
a. How fast it takes you to get to work as a function of how fast you drive.
b. Probability of getting into a car accident as a function of how fast you drive.
c. Your height as a function of age (from age 0 to 100)

Solution Preview

1. Determine whether each of the following is a function or not.
(a) f(x) = 1 if x>1
= 0 otherwise
This is a function because it maps all the points in the domain (all real values of x) to some point in the co-domain (0 or 1)

(b) f(x) = 2 if x>0
= -2 if x<0
= 2 or -2 if x = 0
= 0 otherwise
This is not a function because the image of x = 0 is not unique, i.e., 0 in domain is mapped to both 2 and -2. Further, the otherwise portion does not make any sense because the entire domain is covered by the first three definitions of f(x).

(c) f(x) = 5/x
This is not a function in real space because x = 0 does not have any image in the co-domain as division by 0 is not allowed. It can be a function in imaginary ...

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