Section 1.2
This section covers the following topics:
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Definition of a relation
Domain and the range of a relation
Definition of a function
Functional notation
Graphs of relations and functions
Relations. Domain and Range of a Relation.
In this class we’ll mostly work with formulas such as 𝑦 = 𝑥 2 − 14𝑥 or 𝑦 = √2𝑥 + 5. The main feature of these formulas is that they
are already solved for y (or can be solved for y). If we know the value of x e.g. if we know that x = 5, 8, 26 etc. we can plug it in and
easily calculate y. For any given x there will be exactly one corresponding y value, so that the pair (x, y) is a solution to our equation.
In these situations we’ll say that y is a function of x and we’ll write 𝑦 = 𝑓(𝑥) to tell everyone that there is a formula for y in terms of
x. 𝑓(𝑥) is the name for that formula. Some equations are more difficult than that. Sometimes, for any given x there may be two or
more solutions for y values. For example, in the equation 𝑦 2 = 𝑥 + 8, for x=1 there are two solutions for y, namely y = 3 and y=-3.
Anytime two variables depend on each other, no matter how, we’ll say that those variables form a relation. The most convenient
way to specify the relation is to write down all pairs (x, y) of the values of x and y that “go together” using set notation e.g.
{(8, 10), (−2,5), (3,7)} or as a table:
x
8
-2
3
y
10
5
7
Definition of a relation:
A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the
set of all second components is called the range of the relation.
A relation can also be given as:
- a table with x and y columns
- a set (list) of ordered pairs
- a symbolic graph with arrows
- a graph in the rectangular system of coordinates
- an equation
1. Find the domain and the range of the relation {(6, −5), (8, −11), (8, −12), (10, −5)}
Definition of a function:
A function is relation where every input x is used only once. In other words, there are no ordered pairs with the same x but different
y values. For every x there is exactly one y-value (but not the other way around, there may be pairs with different x but the same y).
𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛: {(5, 0), (7, 6), (9, 6)}
𝑁𝑂𝑇 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛: {(5,0), (5,6), (9,11)}
2. Determine which of the following relations are function(s). Circle the functions.
a) {(0,5), (3,9), (5,9)}
b) {(−5,8), (88,7), (−5,9)}
c) {(10,15), (13,1), (25, −2)}
Functions as equations
The solutions (x, y) of an equation represent a relation. For example, equation 𝑥 4 = 𝑦 4 determines a relation. Some of the solutions are (2, 2),
(2, -2), (-2, 2), (-2,-2) etc. This relation is not a function, since for x=2 there are two corresponding y-values, y=2, and y=-2. An equation
determines y as a function of x, if it can be solved for y, and the solution is unique. For example, the equation 2𝑥 − 3𝑦 = 6 determines a
2
function, because it can be solved uniquely for y, namely 𝑦 = 3 𝑥 − 2. On the other hand, equation 𝑦 2 = 𝑥 + 3 doesn’t define y as a function of
x because the solution for y is 𝑦 = ±√𝑥 + 3. So for most x there are two y-values, e.g. for x=6, the y=3 or y=-3.
Here is another way to say when an equation specifies a function or not:
If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a
function of x.
A rule of thumb: an equation that uses an even power of y i.e. 𝑦 2 , 𝑦 4 , 𝑦 6 , 𝑒𝑡𝑐. typically does not represent a function
3. Determine whether the equation defines y as a function of x:
a) 2𝑦 + 3𝑥 2 = 0
b)
2𝑦 2 − 4𝑥 2 = 5
Functional Notation
𝑥+2
3
In this class we’ll be working with equations/formulas looking like 𝑦 = 𝑥−3 or 𝑦 = −5𝑥 6 + √𝑥. In those formulas the variable x
inside the formula is called the independent variable, and the variable y is called the dependent variable. As the formulas for y get
more complicated it is convenient to use special notation to keep things manageable.
As an example, consider the equation: 𝑦 = 0.014𝑥 2 − 0.24𝑥 + 8.8. In functional notation, we give an name to the formula for y,
e.g. 𝑓(𝑥), and we write 𝑦 = 𝑓(𝑥), 𝑓(𝑥) = 0.014𝑥 2 − 0.24𝑥 + 8.8. Every time we write 𝑓(𝑥), we mean: 0.014𝑥 2 − 0.24𝑥 + 8.8.
In other words, 𝑓(𝑥) replaces the longer formula: 0.014𝑥 2 − 0.24𝑥 + 8.8. In this notation 𝑓(𝑥) is the output of the function i.e. the
same as y.
Figure 1. Example of functional notation.
4. Evaluate the function at the given value of the independent variable and simplify.
𝑥+3
a) 𝑔(𝑥) = 𝑥−4 , 𝑔(−3), 𝑔(8)
b) ℎ(𝑥) = √𝑥 − 6 + 5𝑥, ℎ(10)
5. Simplify ℎ(𝑚 + 3) , for ℎ(𝑥) = −2𝑥 2 + 3.
Reading values of a function from a graph
6. A function 𝑦 = 𝑓(𝑥) is graphed below. Use the graph to answer the following questions:
a) find 𝑓(2), 𝑓(−4)
b) find all x such that 𝑓(𝑥) = 3
7. The following is the graph of the T Cell Count y, as a function of the number of years after infection x. What is 𝑓(5)? For what
value of x is 𝑓(𝑥) = 100?
The Vertical Line Test for Functions
Recall that we can graph any relation regardless of whether it’s a function or not. If a relation is not a function then there will be at
least two points on the graph with same x coordinate. Those points lie on a vertical line. This vertical line will intersect the graph at
least two times. We’ll say that a graph fails the Vertical Line Test if there is a vertical line intersecting the graph at least two times.
When that happens, the graph does not represent a function. On the other hand, if every vertical line intersects the graph at most
once we’ll say that the graph passes the Vertical Line Test. In that case, the graph does represent a function.
In other words:
If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.
8. Use the Vertical Line Test to identify graphs in which y is a function of x.
Identifying the Domain and Range of a Function from its Graph
Example 1
Example 2
9. Find the domain and the range of the relations graphed below:
a)
b)
c)
More on Obtaining Information from Graphs. Mixed Problems.
10. A large tank is being filled with water. The amount of water (in gallons) can be represented by 𝑉(𝑡) = 5500𝑡, where t is the time in hours.
a) Find V(1) and V(3.5).
b) Interpret the meaning of the function values found in part (a) (i.e. describe them in words).