Question

what are functions? functions
output. functions are rules that state every input has exactly one
mappings.
functions can be found with charts/graphs, tables, words, equations, or
looking at the coordinate plane, inputs are the ______ and outputs are the
____. the inputs are referred to as the ____, and
the outputs are referred to as the______.
coordinate plane grid with domain and range labels
a common example of a function is the ______.
how many birthday’s do you have? each person has exactly______
birthday. therefore, a birthday is a function.
function notation
function notation box: ( y = f(x) ) with labels output, rule, input
this is read ______.
the rule affects how the input creates the output.
piecewise functions – in the function, there are different ______ for
different ______.

Explanation:

1. First blank: Functions are rules that assign each input to exactly one output, so the first blank (describing what functions are) should be "Functions".
2. Inputs in coordinate plane: In a coordinate plane, the input values for a function (when thinking of \( y = f(x) \)) are the \( x \)-values (horizontal axis), and the outputs are the \( y \)-values (vertical axis). So the second blank (inputs) is "x - values" and the third blank (outputs) is "y - values".
3. Inputs and outputs terminology: The set of all input values (x - values) is called the "Domain", and the set of all output values (y - values) is called the "Range". So the fourth blank (inputs' term) is "Domain" and the fifth blank (outputs' term) is "Range".
4. Common function example: A common example of a function is something like "relationship between time and distance" (or more simply, a linear function like \( y = 2x \), but a relatable example is "birthday - person" as given later, or a more mathematical one like "the area of a square as a function of its side length"). But from the context of the birthday example, a common function example could be "the relationship between a person and their birthday" (but more generally, a function like \( y=x + 1 \) or a real - world function like "cost as a function of number of items bought").
5. Number of birthdays: Each person has exactly "one" birthday (since a person is born on one day, so for the input "person", the output "birthday" is unique), so the blank here is "one".
6. Function notation reading: The function notation \( y = f(x) \) is read as "y equals f of x".
7. Piecewise functions: In piecewise functions, there are different "rules" (or "formulas") for different "intervals" (or "domains", i.e., different sets of input values). So the blanks for piecewise functions are "rules" (or "formulas") and "intervals" (or "domains").

Answer:

1. Functions
2. x - values; y - values
3. Domain; Range
4. (Example: relationship between time and distance / \( y = 2x \) / etc. - a function example)
5. one
6. y equals f of x
7. rules (formulas); intervals (domains)