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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 (xvalues) 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 (xvalue). Its value determines the value of the dependent variable. Graphed on the xaxis.
 Interval Notation – A shorthand way of writing intervals using parentheses and brackets.
 Range – The set of all possible output (yvalues) 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 xvalues (horizontal) and y are the yvalues (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 xvalues and C are the yvalues
 Example in Words
 The amount of time it takes to finish a race might be a function of the distance of the race

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 yvalues on the graph
 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 yvalues below zero on the graph. All of the yvalues on this graph start at zero and go up.
 Also written in Interval Notation like this:
 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:
 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 yvalue
 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 yvalue for each xvalue
 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) = x^{2} + 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) = 5^{2} + 2(5)
 f(5) = 25 + 10
 f(5) = 35
 To find f(a), substitute a into every input (xvalue)
 To find f(a + 3), substitute (a + 3) into every input (xvalue)
 f(a + 3) = (a + 3)^{2} + 2(a + 3)

Interval Notation 

Important!
Practice: Apex Study 2.1
 Practice: Pgs 2, 4, 9, 10, 14
 Watch animations on Pgs 3
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