**Section 7: Formulas and Functions
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A formula is a mathematical rule for doing a specific calculation. It's generally stated in symbolic and equation form.

** Example 24 ***d = rt is used to express "distance equals rate times time".*

**Example 25** *P = 21 + 2w is used to express "Perimeter of a rectangle equals two times length plus two times width".*

Example 26*i= Prt tells how much simple interest a principle investment will earn in a given time.*

You use a formula by replacing the variable(s) with the value(s) you know and calculating the value(s) for the variable(s) you don't know. Consider the perimeter formula. If I know the length is 5 feet and the width is 3 feet, I can substitute these into the formula and get P = 2(5) + 2(3). Then, following the order of operations, I can calculate the perimeter to be 16 feet.

Sometimes, however, a formula isn't in the form we want. Consider the perimeter formula again. Perhaps I know the perimeter is 20 feet and the width is 3 feet. Then I would want to know the length. When I substitute the known values into the formula I get 20 = 21+2(3). Using algebra, I can "solve" this for I. Another option would have been to "solve" the original formula for 1 and get a new formula. The next section covers the algebraic skills needed to do this.

A **function** is a particular type of formula. The main idea to keep in mind is "input" and "output". A function is a formula with a particular input variable (or variables) and particular output variable (or variables). Generically speaking, *y* is the output, *x* is the input, and *f* is the function. When you see *y = f(x)* that means that *x* is input into a function formula *f *and *y* is the output. In words we say "*y* equals *f* of *x*". For example if *f(x) = *2*x* +5 then *y = f(x) *means that we take any input *x*, multiply it by 2, add 5 to the answer, and get a *y* output. In this case *f*(3) is 2(3) +5 or 11. Then *y* = 11.

Although most of the functions in basic probability and statistics will only have a single input in their formulas, other functions can have multiple inputs. For example, the perimeter formula used before is a function with two inputs, *l* and *w*. We could write it *Perimeter = P(l, w) = 21 +2w*.

An important rule about functions is that a function ** cannot** get different outputs for a single input or set of inputs. Let's stick with the perimeter example. If I use 5 and 4 as inputs, P(5,3) = 2(5) +2(4) = 10 +8 = 18. If I try it again tomorrow with 5 and 4 again, I'll get 18 again. The numbers 5 and 4 make a single set of inputs, so if this is a function, they cannot give me a different output. Notice that this is a one way rule. There is not a similar restriction on outputs. **A single output **can be obtained by **different inputs**. For example P(6,3) = 2(6) + 2(3) is also equal to 18.

This is a subtle point and sometimes confusing. Look at bold words in the previous paragraph. Different outputs with single input is a no-no. A single output with different inputs is OK.