2.1 – What Is a Function?

Objectives

• Distinguish between relations and functions.
• Calculate domain and range of functions algebraically.
• Identify domain and range of functions graphically.

Key Terms

• Dependent Variable – A variable in a function whose value is determined by the value of the independent variable.
• Domain – The set of all possible input (x-values) of the independent variable.
• Function – A special type of relationship in algebra where two quantities are related to each other so that one quantity depends on (or is a function of) the other. For each input number, there is exactly one output number.
• Independent Variable – The input variable in a function (x-value). Its value determines the value of the dependent variable. Graphed on the x-axis.
• Interval Notation – A shorthand way of writing intervals using parentheses and brackets.
• Range – The set of all possible output (y-values) of the dependent variable. The difference between the maximum value and the minimum value of a data set, written as range = maximum − minimum.
• Relation – A set of ordered pairs [(x,y) pairs]. x is the domain / input / independent variable. y is the range / output / dependent variable. If each element in the domain is paired with just one element in the range, the relation is also a function. A relation is sometimes called a mapping.

Notes

Naming Functions
• Naming Functions: Standard Form: y = f(x)
• How to write it: “y is a function of x” is written as y = f(x).
• How to say it: the algebraic expression says “y equals f of x.”
• What it means:
• y depends on x
• The quantity y is a function of the quantity x
• x is input (independent) and y is output (dependent)
• Each value of x will result in exactly one value of y.
• Function Name: f is the function name
• This function is the rule for converting x into y
• Graphed: x are the x-values (horizontal) and y are the y-values (vertical)

• Example (with other variables)
• Write it: “C is a function of t” is written algebraically as “C = g(t)”
• Say it: “C equals g of t”
• Means:
• C depends on t
• The quantity C is a function of the quantity t
• t is the input (independent) and C is the output (dependent)
• Each value of t will result in exactly one value of C.
• Function Name: g is the function name
• This function is the rule for converting t into C
• Graphed: t are the x-values and C are the y-values

• Example in Words
• The amount of time it takes to finish a race might be a function of the distance of the race
Domain
• Domains (Graphed)
• To find the domain, look at the drawing on the graph.
• Notice how far LEFT and RIGHT the graph can go.
• All the x-values on the graph
• Domain Limits – Sometimes, the x-values have limits.
• Explicit (direct) limits are given to you:
• You were given that x must be greater than or equal to zero in this example.
• Inferred (indirect) limits are those in which you must think about algebraic rules you already know:
• Ask yourself: Can x by any number?  Are there any limits to having a variable in the denominator?
• In this example, x cannot be 4 because, if you substitute 4 for x, then you get 4 – 4 = 0 and you cannot have 0 in the denominator of a fraction, ever!
• You should write this domain in Interval Notation like this:
• See more about Interval Notation below
• Another inferred (indirect) example of limits with a radical
• Ask yourself: Are there any limits to having a variable inside a square root (radical)?
• In this example, x can be equal to -2 (because the square root of zero is zero); however, x cannot be less than -2 because you cannot have a negative value inside a square root (it would be an imaginary number).
• You should write this domain in Interview Notation like this:
• Graph

Range
• Range (Graphed)
• To find the range, look at the drawing on the graph.  Notice how far UP and DOWN the graph can go.
• All the y-values on the graph

• Graph Example 1

• For this graph:
• Domain: All real numbers
• … because the graph goes to the left infinitely and to the right infinitely.
• Also written in Interval Notation like this:
• Range: y ≥ 0;  y is greater than or equal to zero
• … because there are no y-values below zero on the graph.  All of the y-values on this graph start at zero and go up.
• Also written in Interval Notation like this:

• Graph Example 2

• For this graph:
• Domain: All real numbers
• … because the graph goes to the left infinitely and to the right infinitely
• Also written in Interval Notation like this:
• Range: All real numbers
• …because the graph goes to the left infinitely and to the right infinitely
• Also written in Interval Notation like this:

• Graph Example 3

• Domain: All real numbers because it goes left and right infinitely
• Range: is the set {3} because it does not go above or below 3

• Graph Example 4: Is the following graph of a relation also a function?

• This graph shows a relation that is NOT a function
• …because the value x = 3 is assigned to more than one y-value
• On this graph, we can see points (3, -3), (3, -2), (3, -1), (3, 0), (3, 1), (3, 2), (3, 3), etc.
• Remember: functions have only 1 y-value for each x-value
• For this relation, when x = 3, y is { … -3, -2, -1, 0, 1, 2, 3 … }
Domain and Range
• Think of domain and range like a machine.
• You put in a domain and you get out a range.
• Ex. Like cooking: you put in the ingredients (domain) and you get out the cake (range)

• Real World Example (see the Baby growth chart below)
• What are your x (input values)?  What are your y (output values)?
• Remember: x is the domain, y is the range

Evaluating a Function
• Example f(x) = x2 + 2x.
• What is the value of this function when x = 5? This is another way of asking, what is the f(5)?
• To find f(5), just substitute 5 in place of each x in the function and calculate the answer.
• f(5) = 52 + 2(5)
• f(5) = 25 + 10
• f(5) = 35
• To find f(a), substitute a into every input (x-value)
• f(a) = a2 + 2(a)
• To find f(a + 3), substitute (a + 3) into every input (x-value)
• f(a + 3) = (a + 3)2 + 2(a + 3)
Interval Notation