Chapter 2.2 Functions

Relations and Functions Recall from Section 2.1 how we described one quantity in terms of another. The letter grade you receive in mathematics course depends on your numerical score.

The amount you pay (in dollars) for gas at the gas station depends on the number of gallons pumped. The dollars spent on entertainment depends on the type of entertainment.

We used ordered pairs to represent these corresponding quantities. For example, (3, $5.25) indicates you pay $5.25 for 3 gallons of gas.

Since the amount you pay depends on the number of gallons pumped, the amount (in dollars) is called the dependent variable and the number of gallons pumped is called the independent variable.

Generalizing, if the value of the variable y depends on the value of the variable x, then y is the dependent variable and x is the independent variable.

Because we can write related quantities using ordered pairs, a set of ordered pairs such as {(3, 5.25), (8, 10), (10, 12.50)} is called a relation.

Relation A relation is a set of ordered pairs.

Function A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.

Note The relation from the beginning of this section representing the number of gallons of gasoline and the corresponding cost is a function such each x- value is paired exactly one y-value.

You would not be happy, for example, if you and a friend each pumped 20 gallons of regular gasoline a the same station and you bill was $32 while his bill was $28.

Example 1 Deciding Whether Relations Define Functions Decide whether each relation defines a function. F = {(1, 2), (-2, 4), (3, -1)}

Example 1 Deciding Whether Relations Define Functions Decide whether each relation defines a function. G = {(1, 1), (1, 2), (2, 3)}

Example 1 Deciding Whether Relations Define Functions Decide whether each relation defines a function. H = {(-4, 1), (-2, 1), (-2, 0)}

In a function there is exactly one value for the dependent variable, the second component, for each value of the independent variable, the first component. This is what makes functions so important in applications.

Relations and functions can also be expressed as a correspondence or mapping from one set to another, as shown in figure 17 for function F and relation H from Example 1.

The arrow from 1 to 2 indicates that the ordered pair (1, 2) belongs to F— each first component is paired with exactly one second component.

In the mapping for relation H, which is not a function, the first component -2 is paired with two different second components, 1 and 0.

x y Since relations and functions are sets of ordered pairs, we can represent them using tables and graphs. y x (-2, 4) (1, 2) (3, -1) Graph of F Figure 18 A table and graph for function F is shown in figure 18.

Finally, we can describe a relation or function using a rule that tells how to determine the dependent variable for a specific value of the independent variable.

The rule may be given in words: for instance, “the dependent variable is twice the independent variable.” Usually the rule is an equation:

Note Another way to think of a function relationship is to think of the independent variable as an input and the dependent variable as an output.

This is illustrated by the input-output (function) machine for the function defined by

Domain and Range For every relation there are two important set of elements called the domain and the range

Domain and Range In a relation, the set of all values of the independent variable (x) is the domain; the set of all values of the dependent variable (y) is the range

Example 2 Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relations defines a function. {(3, -1), (4, 2 ), (4, 5), (8, 8)}

Example 2 Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relations defines a function ABCABC (b)

Example 2 Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relations defines a function. x y (c)

Example 3 Finding Domains and Ranges of Relations Give the domain and range of each relation. (-1, 1) (1, 2) (4, -3) (0, -1) (a)

Domain Range Example 3 Finding Domains and Ranges of Relations (b) Give the domain and range of each relation.

(c) Example 3 Finding Domains and Ranges of Relations Give the domain and range of each relation.

(d) Example 3 Finding Domains and Ranges of Relations Give the domain and range of each relation.

Since relations are defined by equations, such as y = 2x + 3 and y 2 = x, we must sometimes determine the domain of a relation from its equation. In this book, we assume the following agreement on the domain of a relation.

To illustrate this agreement, since any real number can be used as a replacement for x in y = 2x + 3, the domain of this function is the set of all real numbers.

As another example, the function defined by y = 1 / x has all real numbers except 0 as domain, since y is undefined if x = 0. In general, the domain of a function defined by an algebraic expression is all real numbers, except those numbers that lead to division by 0 or an even root of a negative number.

Determining Functions from Graphs or Equations Most of the relations we have seen in the examples are functions—that is, each x-value corresponds to exactly one y-value. Since each value of x leads to only one value of y in a function, any vertical line drawn through the graph of a function must intersect the graph in at most one point. This is the vertical line test for a function

The graph in Figure 19(a) represents a function

Example 4 Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function. (-1, 1) (1, 2) (4, -3) (0, -1) (a)

Domain Range Example 4 Using the Vertical Line Test (b) Use the vertical line test to determine whether each relation is a function.

(c) Example 4 Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function.

(d) Example 4 Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function.

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range. y = x + 4

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range.

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range.

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range. y 2 = x

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range.

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range.

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range.

Example 5 Identifying Functions Domains and Ranges From Equations Decide whether each relation defines a function and give the domain and range.

Function Notation When a function f is defined with a rule or an equation using x and y for the independent and dependent variables, we say “y is a function of x” to emphasize that y depends on x.

We use the notation y = f(x) called function notation, to express this and read f(x) as “f of x.”

For example if y = f(x) = 9x – 5 and x = 2, then we find y, or f(2), by replacing x with 2. y = f(x) = 9x - 5 = 9(2) – 5 = 18 – 5 = 13

if y = f(x) = 9x - 5 f(0) =

if y = f(x) = 9x - 5 f(-3) =

Example 6 Using Function Notation Let f(x) = -x 2 + 5x -3 Find f(2)

Example 6 Using Function Notation Let f(x) = -x 2 + 5x -3 Find f(q)

Example 7 Using Function Notation Let g(x) = 2x + 3 Find g(a+1)

Example 8 Using Function Notation Let f(x) = 3x - 7 Find f(3)

Example 8 Using Function Notation f = {(-3,5 ), (0,3), (3,1), (6,-1), Find f(3)

Example 8 Using Function Notation Find f(3)

Example 9 Using Function Notation Find f(3)

Example 9 Using Function Notation Find f(3)

Example 9 Wrting Equations Using Function Notation Rewrite each equation using function notation. Then find f(-2) and find f(a) y = x 2 + 1

Example 9 Wrting Equations Using Function Notation Rewrite each equation using function notation. Then find f(-2) and find f(a) x – 4y = 5

Increasing, Decreasing, and Constant Functions Informally speaking, a function increases on an interval of its domain if its graph rises from left to right on the interval. It decreases on an interval of its domain if its graph falls from left to right on the interval. It is constant on an interval of its domain if its graph is horizontal on the interval.

Example 2 Finding Domains and Ranges of Relations In Figure 24 the function increases on the interval [-2, 1] because the y-values continue to get larger in that interval.

Example 2 Finding Domains and Ranges of Relations It is constant on the interval [1, 4] because the y- values are always 5 for all x-values there.

Example 2 Finding Domains and Ranges of Relations In Figure 24 the function decreases on the interval [4, 6] because the y-values continue to get smaller in that interval.

Example 10 Determinng Intervals over Which a Function is Increasing, Decreasing, or Constant Determine the intervals over which a function is increasing, decreasing, or constant

Example 11 Interpreting a Graph Figure 27 shows the relationship between the number of gallons of water in a small swimming pool and the time in hours. By looking at this graph of the function, we can answer questions about the water level in the pool at various times.

Example 11 Interpreting a Graph For example, at the time 0 the pool is empty.

Example 11 Interpreting a Graph The water level then increases, stays constant for a while, decreases, then becomes constant again.

Example 11 Interpreting a Graph What is the maximum number of gallons of water in the pool? When is the maximum water level first reached?

Example 11 Interpreting a Graph For how long is the water level increasing? decreasing? constant?

Example 11 Interpreting a Graph How many gallons of water are in the pool after 90 hours?

Example 11 Interpreting a Graph Describe a series of events that could account for the water level changes shown in the graph?