What is a function? How do we define functions?

According to the American Heritage Dictionary a function is

1. The action for which one is particularly fitted or employed.
2. a. Assigned duty or activity. b. A specific occupation or role: in my function as chief editor.
3. An official ceremony or a formal social occasion.
4. Something closely related to another thing and dependent on it for its existence, value, or significance: Growth is a function of nutrition.
5. Abbr. f Mathematics. a. A variable so related to another that for each value assumed by one there is a value determined for the other. b. A rule of correspondence between two sets such that there is a unique element in the second set assigned to each element in the first set.

Examples: social security numbers (<-> people), income tax (domain= taxable income), sales tax (domain= taxable purchase), shoe size (domain = foot length), population (domain = time span), area of a field (domain = length-width pair), area or circumference of a circle (domain = radius or diameter of circle), length of side of square (domain = area of square)

How do we define functions? In the case of functions like the people-social security number function, the function is defined by means of a table that lists the person's name and the assigned number. (Note that telephone numbers are not a function of people because one person can have several phone numbers but we can have a function that assigns each household to a phone number—the phone company does this when it creates the bill for a phone number). Sales tax is given by means of a table and there are income tax tables.

We can define a function by giving a specific rule. The rule can be simple: the area of a square is the square of the length of one side (A = s²); area of a circle is A = Pi(r²). It can also be complex: Federal income tax (for some people): for single people the rule states if taxable income (TI) is over $0 but not over $24,650, tax is 15% of TI; if TI is over $24,650 but not over $59,750 tax is $3,697.50 + 28% of amount over $24,650; if TI is over $59,750 but not over $124,650, tax is $13,525.50 + 31% of amount over $59,750; if TI is over $124,650 but not over $271,050, tax is $33,644.50 + 36% of amount over $124,650; if TI is over $271,050, tax is $86,348.50 + 39.6% of amount over $271,050. Income tax is actually determined by a combination of tables (TI under $100,000) and rules (TI at least $100,000).

We can define a function by drawing a graph. Electrocardiograms and EEGs are functions of time that are given as graphs; temperature during the day, height of the weight on a spring (as a function of time) can be shown with graphs.

We can use function notation to give the function a name and to state clearly what the independent variable will be. For example for the area of a square we can write A(s) = s². A woman's shoe size is given by the rule s(x) = 3x - 21, where x is the length of the woman's foot. We could also write s(#) ) = 3# - 21 or shoe(w) = 3w - 21.

There is a problem with the shoe function. Shoes do not come in an infinite number of sizes. Women's shoes come in half sizes beginning with a size 5. What length foot should fit in a size 5 shoe? What would you have to do if your foot is smaller than that? What would be a reasonable domain for the shoe function? What is the corresponding range?

The shoe function, sales tax function, and income tax functions are all examples of functions with discrete variables. A variable is discrete if distinct values are separated from one another by intervals. In the case of the shoe function, the shoe size, which is the dependent variable, is a discrete variable. It comes only in half units: 5, 51/2, 6, 61/2, etc. In the case of the sales tax and the income tax functions both of the variables are discrete: a penny is the smallest unit of money that we can use. In the case of the shoe function the length of the foot varies continuously between about 8 and 12 inches. The only limitation is the accuracy of our measurement. Another function which has a continuous independent variable and a discrete dependent variable is the postage function which assigns the cost of the postage to the weight of a letter (if the letter weighs more than 0 ounces but not more than 1 ounce a stamp costs 32˘, for each additional ounce you add 23˘ in postage). We could write

             32   for  0 < w <= 1
stamp(w) =   55   for  1 < w <= 2
             78   for  2 < w <= 3
             etc.

 where  w  is the weight of the letter

This is an example of a step function (as is the true shoe function)

Problem 1. Quad is defined by this formula: Quad(#) = 100 + 21# – #^2
Find each of the following, simplifying where possible:

1. Quad(-5)
2. Quad(-x)
3. Quad(z+5)
4. Quad(z) + 5
5. 1 - Quad(t)
6. Quad(1 - t)
7. Quad(1/x)
8. 1/Quad(x)
9. the values of # for which Quad(#) equals zero

Problem 2. Find the domain and range of each function:

1. y = 10 - 2x
2. Q(t) = 10 - t²
3. R(t) = 10 - 5t²
4. T(x) = 5/x
5. U(x) =(x + 5)^1/2
6. V(x) = x^1/2 + 5
7. red(v) = 1/(v - 2)
8. blue(v) = (1/v) - 2

Problem 3. What are the graphs for the following functions:

1. f(x) = x
2. g(x) = |x|
3. s(x) = x²
4. c(x) = x³
5. r(x) = x^1/2
6. rat(x) = 1/x