11 table 2.1 defines a correspondence between a set of percent scores and a set of letter grades....

11

Table 2.1 defines a correspondence between a set of percent scores and a set of letter grades.

For each score from 0 to 100, there corresponds only one letter grade.

Relations

Table 2.1

22

The score 94% corresponds to the letter grade of A. Using ordered-pair notation, we record this correspondence as (94, A).

The equation d = 16t

2 indicates that the distance d that a rock falls (neglecting air resistance) corresponds to the time t that it has been falling.

For each nonnegative value t, the equation assigns only one value for the distance d.

Relations

33

According to the equation, in 3 seconds a rock will fall 144 feet, which we record as (3, 144).

Some of the other ordered pairs determined by d = 16t

2

are (0, 0), (1, 16), (2, 64), and (2.5, 100).

Equation: d = 16t

2

If t = 3, then d = 16(3)2 = 144

Relations

44

The graph in Figure 2.19, defines a correspondence between the length of a pendulum and the time it takes the pendulum to complete one oscillation.

Relations

Figure 2.19

55

For each nonnegative pendulum length, the graph yields only one time.

According to the graph, a pendulum length of 2 feet yields an oscillation time of 1.6 seconds and a pendulum length of 4 feet yields an oscillation time of 2.2 seconds, where the time is measured to the nearest tenth of a second.

These results can be recorded as the ordered pairs (2, 1.6)

and (4, 2.2).

Relations

66

The table 2.1, earlier equation and the following graph each determine a special type of relation called a function.

Functions

Table 2.1

77

Definition of a Function

A function is a set of ordered pairs in which no two ordered pairs have the same first coordinate and different second coordinates.

Although every function is a relation, not every relation is a function.

For instance, consider (94, A) from the grading correspondence. The first coordinate, 94, is paired with a second coordinate, A.

Functions

88

Functions may have ordered pairs with the same second coordinate.

For instance, (94, A) and (95, A) are both ordered pairs that belong to the function defined by Table 2.1.

Thus a function may havedifferent first coordinates andthe same second coordinate.

Functions

Table 2.1

99

The equation d = 16t

2 represents a function because for each value of t there is only one value of d.

However, not every equation represents a function. For instance, y2 = 25 – x2 does not represent a function.

The ordered pairs (–3, 4) and (–3, –4) are both solutions of the equation.

However, these ordered pairs do not satisfy the definition of a function: There are two ordered pairs with the same first coordinate but different second coordinates.

Functions

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The domain of a function is the set of all the first coordinates of the ordered pairs. The range of a function is the set of all the second coordinates.

In the function determined by the grading correspondence in Table 2.1, the domain is the interval [0, 100]. The range is {A, B, C, D, F}.

In a function, each domain elementis paired with one and only onerange element.

Functions

Table 2.1

1111

If a function is defined by an equation, the variable that represents elements of the domain is the independent variable. The variable that represents elements of the range is the dependent variable.

For the situation involving the free fall of a rock, we used the equation d = 16t

2.

The elements of the domain represented the time the rock fell, and the elements of the range represented the distance the rock fell. Thus, in the d = 16t

2, the independent variable is t and the dependent variable is d.

The specific letters used for the independent and dependent variables are not important.

Functions

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State whether the relation defines y as a function of x.

a. {(2, 3), (4, 1), (4, 5)}

b. 3x + y = 1

c. –4x2 + y2 = 9

d. The correspondence between the x values and the y values in Figure 2.20.

Example 1 – Identify Functions

Figure 2.20

1313

Functions can be named by using a letter or a combination

of letters, such as f, g, A, log, or tan.

If x is an element of the domain of f, then f (x), which is read

“f of x” or “the value of f at x,” is the element in the range of

f that corresponds to the domain element x.

Function Notation

1414

Let f (x) = x2 – 1, and evaluate.

a. f (–5)

b. f (3b)

c. 3f (b)

d. f (a + 3)

e. f (a) + f (3)

Example 2 – Evaluate Functions

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Piecewise-defined functions are functions that are represented by more than one expression.

For instance, the function f defined below consists of three pieces, 2x + 1, x2 – 1, and 4 – x.

Function Notation

1616

To evaluate the function at x, determine the interval in which x lies and then use the expression that corresponds to that interval to evaluate the function.

The following examples show how to evaluate f (–4), f (5),

and f (2).

Since –4 < –2, use 2x + 1. f (–4) = 2(–4) + 1 = –7

Since 5 > 3, use 4 – x. f (5) = 4 – 5 = –1

Since –2 < 2 < 3, use x2 – 1. f (2) = 22 – 1 = 3

Function Notation

1717

Sometimes the domain of a function is stated explicitly.

For example, each of f, g, and h below is given by an equation followed by a statement that indicates the domain.

f (x) = x2, x > 0

h(x) = x2, x = 1, 2, 3

Although f and h have the same equation, they are different functions because they have different domains.

Function Notation

1818

If the domain of a function is not explicitly stated, then its domain is determined by the following convention.

Domain of a Function

Unless otherwise stated, the domain of a function is the set of all real numbers for which the function makes sense and yields real numbers.

Function Notation

1919

If a is an element of the domain of a function f, then

(a, f (a)) is an ordered pair that belongs to that function.

Definition of the Graph of a Function

The graph of a function is the graph of all ordered pairs

that belong to the function.

Graphs of Functions

2020

Graph each of the following. State the domain of each

function.

a. f (x) = 2x – 3

b. g(x) = 2x2 – 3

c. h(x) = – 3

Solution:

For each part, we create a table of ordered pairs for the function, plot the ordered pairs and then draw a graph through them. Although each of the functions has a similar look, their graphs are quite different.

Example 5 – Graph a Function by Plotting Points

2121

It may be that, for a given number b, there is no number in the domain of f for which f (a) = b.

For instance, suppose f (x) = x2 + 3 and we are asked to find a value in the domain of f for which f (a) = 2.

f (a) = 3

a2 + 3 = 2

a2 = –1

a = i

The values of a are complex numbers and not in the domain of f.

Graphs of Functions

Replace f (a) with a2 + 3.

Solve for a.

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Note from the graph in Figure 2.26 that the horizontal line through (0, 2) does not intersect the graph.

Graphs of Functions

Figure 2.26

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A problem of special interest is determining the values in the domain of a function f for which f (a) = 0.

A value a in the domain of a function f for which f (a) = 0 is called a zero of f.

For instance, let f (x) = 2x – 4. When x is 2, we have f (2) = 2(2) – 4 = 0.

Because, f (2) = 0, 2 is a zero of f.

Graphs of Functions

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Here is another example: Suppose g(x) = x2 + 2x – 15. Then

g(x) = x2 + 2x – 15 g(x) = x2 + 2x – 15

g(3) = 32 + 2(3) – 15 g(–5) = (–5)2 + 2(–5) – 15

g(3) = 0 g(–5) = 0

3 is a zero of g. –5 is a zero of g.

In this case, there are two values in the domain of g, –5 and 3, for which f (x) = 0.

Real Zeros and x-Intercepts Theorem

The real number c is a zero of f if and only if (c, 0) is an x-intercept of the graph of y = f (x).

Graphs of Functions

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The definition of a function as a set of ordered pairs in which no two ordered pairs that have the same first coordinate have different second coordinates implies that any vertical line intersects the graph of a function at no more than one point.

This is known as the vertical line test.

The Vertical Line Test for Functions

A graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.

Graphs of Functions

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Consider the graph in Figure 2.28. As a point on the graph moves from left to right, this graph falls for values of x –2, remains the same height from x = –2 to x = 2, and rises for x 2.

The function represented by the

graph is said to be decreasing

on the interval constant

on the interval [–2, 2], and

increasing on the interval

Graphs of Functions

Figure 2.28

2727

Definition of Increasing, Decreasing, and Constant Functions

If a and b are elements of an interval I that is a subset of the domain of a function f, then

f is increasing on I if f (a) < f (b) whenever a < b.

f is decreasing on I if f (a) > f (b) whenever a < b.

f is constant on I if f (a) = f (b) for all a and b.

Graphs of Functions

2828

Recall that a function is a relation in which no two ordered pairs that have the same first coordinate have different second coordinates.

This means that, given any x, there is only one y that can be paired with that x .

A one-to-one function satisfies the additional condition that, given any y, there is only one x that can be paired with that given y.

Graphs of Functions

2929

In a manner similar to applying the vertical line test, we can apply a horizontal line test to identify one-to-one functions.

Horizontal Line Test for a One-To-One Function

If every horizontal line intersects the graph of a function at most once, then the graph is the graph of a one-to-one function.

Graphs of Functions

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For example, some horizontal lines intersect the graph in Figure 2.29 at more than one point.

This is not the graph of a one-to-one function.

Graphs of Functions

Figure 2.29

Some horizontal lines intersect thisgraph at more than one point. This isnot the graph of a one-to-one function.

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Every horizontal line intersects the graph in Figure 2.30 at most once.

This is the graph of a one-to-one function.

Graphs of Functions

Figure 2.30

Every horizontal line intersects thisgraph at most once. This is the graphof a one-to-one function.

3232

The graphs of some functions do not have any breaks or gaps. These functions, whose graphs can be drawn without lifting a pencil off the paper, are called continuous functions.

The graphs of other functions do have breaks or discontinuities. One such function is the greatest integer function or floor function. This function is denoted by symbols such as and int(x).

The value of the greatest integer function at x is the greatest integer that is less than or equal to x.

Greatest Integer Function (Floor Function)

3333

For instance,

To graph the floor function, first observe that the value of the floor function is constant between any two consecutive integers.

For instance, between the integers 1 and 2, we have

int(1.1) = 1 int(1.35) = 1 int(1.872) = 1 int(1.999) = 1

Between –3 and –2 we have

int(–2.98) = –3 int(–2.4) = –3 int(–2.35) = – 3 int(–2.01) = –3


3434

Using the property of the floor function, we can create a table of values and then graph the floor function. See Figure 2.31.


Figure 2.31

3535

The graph of the floor function has discontinuities (breaks) whenever x is an integer.

The domain of the floor function is the set of real numbers; the range is the set of integers.

Because the graph appears to be a series of steps, sometimes the floor function is referred to as a step function.


3636

One application of the floor function is rounding numbers. For instance, suppose a credit card company charges 1.5% interest on an unpaid monthly balance of $237.84.

Then the interest charge I is

I = 237.84 0.015 = 3.5676

Thus the interest charge is $3.57.

Note that the result was rounded to the nearest cent, or hundredth.


3737

The computer program that determines the interest charge uses the floor function to calculate the rounding.

To round a number N to the nearest kth decimal place, we use the following formula.

Here is the calculation for the interest owed.


3838

In this case, N = 237.84 0.015 and k = 2 (round to the second decimal place).


3939

The cost of parking in a garage is $3 for the first hour or any part of the first hour and $2 for each additional hour or any part of an hour thereafter. If x is the time in hours that you park your car, then the cost is given by

C(x) = 3 – 2 int(1 – x), x > 0

a. Evaluate C(2) and C(2.5).

b. Graph y = C(x) for 0 < x 5.

Example 9 – Use the Greatest Integer Function to Model Expenses

4040

To graph C(x) for 0 < x 5, consider the value of int(1 – x) for each of the intervals 0 < x 1, 1< x 2, 2 < x 3, 3 < x 4, and 4 < x 5.

For instance, when 0 < x 1, 0 1 – x < 1.

Thus int(1 – x) = 0 when 0 < x 1.

Now consider 1< x 2. When 1< x 2 , –1 1 – x < 0.

Thus int(1 – x) = –1 when 1< x 2.

Example 9(b) – Solution cont’d

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Applying the same reasoning to each of the other intervals gives the following table of values and the graph of C shown in Figure 2.34.

Example 9(b) – Solution cont’d

Figure 2.34

4242

A car was purchased for $16,500. Assuming that the car depreciates at a constant rate of $2200 per year (straight-line depreciation) for the first 7 years, write the value of v the car as a function of time, and calculate the value of the car 3 years after purchase.

Solution:

Example 10 – Solve an Application

11 table 2.1 defines a correspondence between a set of percent scores and a set of letter grades....

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