1 Functions and Models

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1 Functions and Models


1.1 Four Ways to Represent a Function




Functions (1 of 7)

Functions arise whenever one quantity depends on another. Consider the following four situations.

A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A  r 2. With each

Table 1

Year Population (millions)

1900 1650

1910 1750

1920 1860

1930 2070

1940 2300

1950 2560

1960 3040

1970 3710

1980 4450

1990 5280

2000 6080

2010 6870

B. The human population of the world P depends on the time t. Table 1 gives estimates of the world population P at time t, for certain years. For instance,

P ≈ 2,560,000,000 when t = 1950

For each value of the time t there is a corresponding value of P, and we say that P is a function of t.

positive number r there is associated one value of A, and we say that A is a function of r.


Functions (2 of 7)

C. The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known.

D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t.

Vertical ground acceleration during the Northridge earthquake

Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 19 94. For a given value of t, the graph provides a corresponding value of a.


Functions (3 of 7)

If f represents the rule that connects A to r in Example A, then we express this in function notation as A = f(r).

A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.

We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function.

The number f(x) is the value of f at x and is read “f of x.”

The range of f is the set of all possible values of f(x) as x varies throughout the domain.


Functions (4 of 7)

A symbol that represents an arbitrary number in the domain of a function f is called an independent variable.

A symbol that represents a number in the range of f is called a dependent variable.

In Example A, for instance, r is the independent variable and A is the dependent variable.

It’s helpful to think of a function as a machine (see Figure 2).

Figure 2

Machine diagram for a function f

If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f(x) according to the rule of the


So we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs.


Functions (5 of 7)

Another way to picture a function is by an arrow diagram as in Figure 3.

Figure 3

Arrow diagram for f

Each arrow connects an element of D to an element of E. The arrow indicates that f(x) is associated with x, f(a) is associated with a, and so on.


Functions (6 of 7)

The most useful method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs

   

x f x, | x D

In other words, the graph of f consists of all points (x, y) in the coordinate plane such that y = f(x) and x is in the domain of f.

The graph of a function f gives us a useful picture of the behavior or “life history” of a function.


Functions (7 of 7)

Since the y-coordinate of any point (x, y) on the graph is y = f(x), we can read the value of f(x) from the graph as being the height of the graph above the point x. (See Figure 4).

Figure 4

Figure 5

The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5.

   

x f x, | x D

If f is a function with domain D, then its graph is the set of ordered pairs


Example 1

The graph of a function f is shown in Figure 6.

(a) Find the values of f(1) and f (5).

(b) What are the domain and range of f ?

Figure 6

(a) We see from Figure 6 that the point (1, 3) lies on the graph of f, so the value of f at 1 is f(1) = 3. (In other words, the point on the graph that lies above x = 1 is 3 units above the x-axis.)

When x = 5, the graph lies about 0.7 unit below the x-axis, so we estimate that f(5) ≈ −0.7.

(b) We see that f(x) is defined when 0  x 7,so the domain of f is the closed interval [0, 7]. Notice that f takes on all values from −2 to 4, so the range of f is

y | 2  y 4

 

2, 4


Representations of Functions


Representations of Functions (1 of 1)

We consider four different ways to represent a function:

verbally (by a description in words)

numerically (by a table of values)

visually (by a graph)

algebraically (by an explicit formula)


Example 4

When you turn on a hot-water faucet that is connected to a hot-water tank, the

temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was

turned on.


The initial temperature of the running water is close to room temperature because the water has been sitting in the pipes.

quickly. In the next phase, T is constant at the temperature of the heated water in the tank.

When the tank is drained, T decreases to the

temperature of the water supply. This enables us to make the rough sketch of T as a function of t shown in

When the water from the hot-water tank starts flowing from the faucet, T increases


Which Rules Define Functions?


Which Rules Define Functions? (1 of 5)

What about curves drawn in the xy-plane? Which curves are graphs of functions?

The following test gives an answer.

The Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

The reason for the truth of the Vertical Line Test can be seen in Figure 13.

(a) This curve represents a (b) This curve doesn’t represent a


Which Rules Define Functions? (2 of 5)

If each vertical line x = a intersects a curve only once, at (a, b), then exactly one function value is defined by f(a) = b.

But if a line x = a intersects the curve twice, at (a, b) and (a, c), then the curve can’t represent a function because a function can’t assign two different values to a.


Which Rules Define Functions? (3 of 5)

For example, the parabola xy2  2 shown in Figure 14(a) is not the graph of a function of x because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x.

Figure 14(a)

2 2 x y


Which Rules Define Functions? (4 of 5)

Notice that the equation xy2  2 implies y2  x 2, so y   x  2.

Thus the upper and lower halves of the parabola are the graphs of the functions

 

2 and

 

  2

f x x g x x [See Figures 14(b) and (c).]

Figure 14(b)

 2

y x

Figure 14(c)

   2

y x


Which Rules Define Functions? (5 of 5)

We observe that if we reverse the roles of x and y, then the equation

 

  2  2

x h y y does define x as a function of y (with y as the independent variable and x as the dependent variable). The graph of the function h is the parabola in Figure 14(a).

Figure 14(a)

2 2 x y


Piecewise Defined Functions


Piecewise Defined Functions (1 of 2)

The functions in the following three examples are defined by different formulas in different parts of their domains. Such functions are called piecewise defined functions.


Example 7

A function f is defined by

 

 2   

1 if 1

if 1

x x

f x x x

Evaluate f(−2), f(−1), and f(0) and sketch the graph.


Remember that a function is a rule. For this particular function the rule is the following:

First look at the value of the input. If it happens that x  1, then the value of f(x) is 1 − x.

On the other hand, if x > −1, then the value of f(x) is x2.

Note that even though two different formulas are used, f is one function, not two.

   

   

 

       

       

Since 2 1, we have 2 1 2 3.

Since 1 1, we have 1 1 1 2.

f f


Example 7 – Solution

If x > −1, then f x

 

x2, so the part of the graph of f that lies to the right of the line x = −1 must coincide with the graph of


y x which is a parabola. This enables us to sketch the graph in Figure 15.

Figure 15

The solid dot indicates that the point (−1, 2) is included on the graph; the open dot indicates that the point (−1, 1) is excluded from the graph.

 

 2   

1 if 1

if 1

x x

f x x x

How do we draw the graph of f ? We observe that if x  1, then f(x) = 1 − x, so the part of the graph of f that lies to the left of the vertical line x = −1 must coincide

with the line y = 1 − x, which has slope −1 and y-intercept 1.


Piecewise Defined Functions (2 of 2)

The next example of a piecewise defined function is the absolute value function. We know that the absolute value of a number a, denoted by a , is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have

 0 for every number

a a

For example, 3 3 3 3 0 0 2 1 2 1


 


      

  

In general, we have  

  

if 0 if 0

a a a

a a a

(Remember that if a is negative, then −a is positive.)


Example 8

Sketch the graph of the absolute value function f x

 

x .


From the preceding discussion we know that  

   if 0 if 0

x x

x x x

Using the same method as in Example 7, we see that the graph of f coincides with the line y = x to the right of the y-axis and coincides with the line y = −x to the left of the y-axis (see Figure 16).

Figure 16


Example 10 (1 of 2)

In Example C at the beginning of this section we considered the cost C(w) of mailing a large envelope with weight w.

In effect, this is a piecewise defined function because, from the table shown in the right side,

w (ounces) C(w) (dollars)

0 < w ≤ 1 1.00

1 < w ≤ 2 1.15

2 < w ≤ 3 1.30

3 < w ≤ 4 1.45

4 < w ≤ 5 1.60

we have

 

 

  

   

  


1.00 if 0 1

1.15 if 1 2

1.30 if 2 3

1.45 if 3 4

w w

C w w

w M


Example 10 (2 of 2)

The graph is shown in Figure 18.

Figure 18

Looking at Figure 18, you can see why a function like the one in Example 10 is called a step function.


Even and Odd Functions


Even and Odd Functions (1 of 3)

If a function f satisfies f(−x) = f(x) for every number x in its domain, then f is called an even function. For instance, the function f x

 

x2 is even because

     



 

f x x x f x

The geometric significance of an even

function is that its graph is symmetric with respect to the y-axis (see Figure 19).

Figure 19

An even function

This means that if we have plotted the graph of f for x  0, we obtain the entire graph simply by reflecting this portion about the y-axis.


Even and Odd Functions (2 of 3)

The graph of an odd function is symmetric about the origin (see Figure 20).

Figure 20

An odd function

If we already have the graph of f for x  0, we can obtain the entire graph by rotating this portion through 180 about the origin.

If f satisfies f (−x) = −f (x) for every number x in its domain, then f is called an odd function. For example, the function f x

 

x3 is odd because

   

  3   3  

 

f x x x f x


Example 11

Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f x

 

x5 x (b) g x

 

 1 x4 (c) h x

 

2x x2



         

 

 

        

     

 

5 5 5

5 5


f x x x x x

x x x x

f x

Therefore f is an odd function.

(b) g

 

x   1

 

x 4  1 x4 g x

 

So g is even.

(c) h

 

x 2

   

  x x 2  2x x2

Since h(−x) ≠ h (x) and h(−x) ≠ −h(x), we conclude that h is neither even nor odd.


Even and Odd Functions (3 of 3)

The graphs of the functions in Example 11 are shown in Figure 21. Notice that the graph of h is symmetric neither about the y-axis nor about the origin.

Figure 21

(a) (b) (c)


Increasing and Decreasing Functions


Increasing and Decreasing Functions (1 of 3)

The graph shown in Figure 22 rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval [a, b], decreasing on [b, c], and increasing again on [c, d ].

Figure 22


Increasing and Decreasing Functions (2 of 3)

Notice that if x1 and x2 are any two numbers between a and b with x1 < x2, then f(x1) < f(x2).

We use this as the defining property of an increasing function.

A function f is called increasing on an interval I if

   

1 2 whenever 1 2 in

f x f x x x I

It is called decreasing on I if

   

1 2 whenever 1 2 in

f x f x x x I


Increasing and Decreasing Functions (3 of 3)

In the definition of an increasing function it is important to realize that the

inequality f(x1) < f(x2) must be satisfied for every pair of numbers x1 and x2 in I with x1 < x2.

You can see from Figure 23 that the function

 


f x x is decreasing on the interval

, 0

and increasing on the interval



Figure 23




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