1.2 FUNCTIONS  OF  REAL  NUMBERS

   The next two sections are about real numbers only. The calculus deals with problems in which one quantity depends on one or more others. For example, the area of a circle depends on its radius. The length of a day depends on both the latitude and the date. The price of an object depends on the supply and the demand. The way in which one quantity depends on one or more others can be described mathematically by a function of one or more variables.

 

DEFINITION

 

A real function of one variable is a set f of ordered pairs of real numbers such that for every real number a one of the following two things happens:

(i) There is exactly one real number b for which the ordered pair(a,b) is a member of f.

   In this case we say that f (a) is defined and we write f(a)=b. The number b is called

   the value of f at a.

(ii) There is no real number b for which the ordered pair (a, b) is a member of f. In this case we say that f(a) is undefined.

 

Thus f(a)=b means that the ordered pair (a,b) is an element of f.

Here is one way to visualize a function. Imagine a black box labeled fas in Figure 1.2.1.

Inside the box there is some apparatus, which we can’t see. On both the left and right sides of the box there is a copy of the real line, called the input line and

 

 

 

 

 

 

 

Figure 1.2.1

 

out put line, respectively. Whenever we point to anumber a on the input line, either one point b will light up on the output line to tell us that f(a)=b, or else nothing will happen, in which case f(a)is undefined.

 

A second way to visualize a function is by drawing its graph. The graph of a real function f of one variable is the set of all points P(x, y)in the plane such that y=f(x).To draw the graph, we plot the value of xon the horizontal, or x-axis and the value of f(x)on the vertical, or y-axis. How can we tell whether a set of points in the plane is the graph of some function? By reading the definition of a function again, we have an answer.

 

A set of points in the plane is the graph of some function fif and only if for each vertical line one of the following happens:

 

(1)Exactly one point on the line belongs to the set.

(2)No point on the line belongs to the set.

 

 A vertical line crossing the x-axis at a point a will meet the set in exactly one point (a,b) if f(a) is defined and f(a) = b, and the line will not meet the set at all if f(a) is undefined. Try this rule out on the sets of points shown in Figure 1.2.2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here are two examples of real functions of one variable. Each function will be described in two ways: the black box approach, where a rule is given for finding the value of the function at each real number, and the graph method, where an equation is given for the graph of the function.

 

EXAMPLE 1    The square function.

       The square function is defined by the rule

                            f(x) = x²

        for each number x, the value of f(a)is fond by squaring a. For instance, the values of f(0), f(2), f(-3), f(r), f(r+1) are

        f(0) = 0,   f(2)=4,   f(-3) = 9,

        f(r) = r²,   f(r+1)=r ² +2r+1.

 

The graph of the square function is the parabola with the equation y=x².

The graph of y=x², with several points marked in, is shown in Figure 1.2.3.

 

EXAMPLE 2  The reciprocal function.

        The reciprocal function g is given by the rule

                              g(x) =_______

g(x)is defined for all nonzero x, but is undefined at x=0. Find the following values if they are defined: g(0),  g(2),g(______),  g(______),  g(r+1).

 

        g(0) is undefined,     g(2) =______,    g(____) = -3

        g(____) = ______,    g(r+1) = _______

 

 

The graph of the reciprocal function has the equation y= 1/x. This equation can also be written in the form xy=1. The graph is shown in Figure 1.2.4.

 

In Examples 1 and 2 we have used the variables xand y in order to describe a function. A variable is a letter which stands for an arbitrary real number; that is, it “varies” over the real line. In the equation y=x², the value of ydepends on the value of x; for this reason we say that xis the independent variable and ythe dependent variable of the equation.

 

 

 

 

 

 

 

 

 

 

 

 

In describing a function, we do not always use xand y; sometimes other variables are more convenient, especially in problems involving several functions. The variable tis often used to denote time.

 

It is important to distinguish between the symbol f and the expression f(x). fby itself stands for a function. f(x)is called a term and stands for the value of the function at x. The need for this distinction is illustrated in the next example.

 

EXAMPLE 3  Let hbe the function given by the rule

                   h(t) = t3+1.

t is a variable, his a function, and h(t)is a term. The following expressions are also terms: h(__) , h(x), h(t3), h(t3)+1, h(t3+ 1), h(x) - h(t), h(t+ Δt),the values are computed by careful substitution.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The graph of h is given by the equation x = t3 + 1. In this equation, tis the independent variable and xis the dependent variable. In Figure 1.2.5, the five points

   h(-1)=0, h(___) = ___,     h(0)=1,  h(__)=___ ,    h(1) = 2

are plotted and the graph is drawn.

 

 

 

 

 

 

 

 

 

 

 

Figure 1.2.5

 

DEFINITION

       The domain of a real function f of one variable is the set of all real numbers x such that f(x) is defined.

        The range of f is the set of all values f(x) where x is in the domain of f.

 

EXAMPLE  1 (Continued) The domain of the square function is the set Rof all real numbers. The range is the interval [0, ∞) of all nonnegative reals.

 

EXAMPLE  2 (Continued) Both the domain and the range of the reciprocal function are equal to the set of all real xsuch that x≠0.

 

          When a function is described by a rule, it is understood that the domain is the set of all real numbers for which the rule is meaningful.

 

EXAMPLE  3 (Continued)  The function hgiven by the rule

                          h(t) = t3 +1

        Has the whole real line as its domain and as its range.

 

EXAMPLE  4 (Continued)  Let fbe the function given by the rule

                             f(x) = __________.

            Thus f(x)is the positive square root of 1-x2. The domain of fis the closed

            interval [-1,1]. The range of f is [0,1].

            For instance,

                   f(-2) is undefined,   f(-1) = 0,    f(0) = 1,

                       f(__) =_______,  f(1) =0 ,   f(2) is undefined.

 

The graph of f is given by the equation y= _______.

The equation can also be written in the form

                   x²+ y² = 1, y≥0.

The graph is just the upper half of the unit semicircle, shown in Figure 1.2.6.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.2.6

 

Sometimes a function is described by explicitly giving its domain in addition to a rule.

 

EXAMPLE  5  Let g be the function whose domain is the closed interval [1,2] with the rule

                                  g(x) = x²

 

       The domain and rule can be written in concise form with an equation and extra inequalities, 

               g(x) = x²,    1≤ x≤2.

Note that   

               g(0) is undefined    g(1)=1

               g(2) = 4     g(3) is undefined.

The graph is described by the formulas

               y=x²        1≤ x ≤2

and is drawn in Figure 1.2.7.

 

Some especially important functions are the constant functions, the identity function, and the absolute value function.

 

A real number is sometimes called a constant. This name is used to emphasize the difference between a fixed real number and a variable.

 

For a given real number c, the function fwith the rule

                        f(x) = c

is called the constant function with value c. It has domain R and range {c}.

 

EXAMPLE  6  The constant function with value 5 is described by the rule

                             f(x) = 5.

     Thus      f(0) = 5, f(-3) = 5,    f(1,000,000) = 5.

The graph (Figure 1.2.8) of the constant function with value 5 is given by the equation y=5.

 

 

 

 

 

 

 

 

 

 

 

EXAMPLE  7  The function f given by the rule

                                f(x) = x

is called the identity function.

The graph (Figure 1.2.9) of the identity function is the straight line with the equation y=x.

                          

 

 

 

 

 

 

 

 

Figure 1.2.9

 

The absolute value function is defined by a rule which is divided into two cases.

 

DEFINITION

             The absolute value function | | is defined by

                     

 

 

The absolute value of xgives the distance between xand 0. It is always positive or zero. For example,

                   |3|=3,   |-3|=3,  |0|=0.

The domain of the absolute value function is the whole real line Rwhile its range is the interval [0, ∞).

The absolute value function can also be described by the rule.

                              |x|=____.

Its graph is given by the equation y= ____. The graph is the V shown in Figure 1.2.10.

 

 

 

 

 

 

 

Figure 1.2.10

If a and b are two points on the real line, then from the definition of |x|we see that

 

 

Thus |a-b|is the difference between the larger and the smaller of the two numbers. In other words, |a-b|is the distance between the points aand b, as illustrated in Figure 1.2.11.

 

 

 

Figure 1.2.11.

 

For example, | 2-5|= 3,  | 4-(-4)|= 8. Here are some useful facts about absolute values.

 

THEOREM  1

Let a and b be real numbers.

(i)  | -a |= | a|.

(ii)  | ab|= | a|· |b|.

(iii) if b ≠0, | a/b|=| a|/|b|.

 

PROOF  We use the equation |x|= _____.

(i)  | -a|=______=______= | a|.

(ii)  |ab|=______=______=______= | a|· |b|.

(iii)  the proof is similar to (ii).

 

Warning  the equation   |a+b|= | a|+ |b|is false in general. For example, 

         |2+(-3)|=1, while | 2|+|(-3)|=5.

 

Function arise in a great variety of situations. Here are some examples.

Geometry:

            πr²= area of a circle of radius r

            4πr²= surface area of a sphere of radius r

           __πr²= volume of a sphere of radius r

           sin θ  = the sine of the angle θ

 

Physics:

            s(t)= distance a particle travels from time 0 to t

            v(t)=velocity of a particle at time t

            a(t)=acceleration of a particle at time t

            p(y)=water pressure at depth ybelow the surface

              C= ___ (F - 32) = Celsius temperature as a

                  Function of Fahrenheit temperature

 

Economics:

                  f(t)= population at time t

                  p(t)=price of a commodity at time t

                  c(x)=cost of x items of a commodity

                  d(p)= demand for a commodity at price p,i.e., the

                        amount which can be sold at price p.

 

Functions of two or more variables can be dealt with in a similar way. Here is the precise definition of a function of two variables.

 

DEFINITION

        A real function of two variables is a set f of ordered triples of real numbers

        such that for every ordered pair of real numbers (a, b) one of the following two

        things occurs:

        (i) there is exactly one real number c for which the ordered triple (a,b,c) is a

           member of f. In this case, f(a,b) is defined and we write:

                             f(a,b) = c

        (ii) there is no real number c for which the ordered triple (a, b, c) is a member

           of f. In this case f(a,b) is called undefined.

 

If f is a real function of two variables, then the value of f(x,y)depends on both the value of xand the value of ywhen f(x,y) is defined.

 

A real function fof two variables can be visualized as a black box with two input lines and one output line, as in Figure 1.2.12.

 

 

 

          

 

 

 

 

Figure 1.2.12

 

The domain of a real function f of two variables is the set of all pairs of real numbers

 (x, y) such that f(x,y)is defined.

 

The most important examples of real functions of two variables are the sum, difference, product, and quotient functions:

                f(x, y) = x +y,      f(x, y) = xy.

                F(x,y) =x - y,      f(x,y) = x/y.

 

The sum, difference, and product functions have the whole plane as domain. The domain of the quotient function is the set of all ordered pairs(x,y) such that y≠0.

 

Here are some examples of functions of two or more variables arising in applications.

 

Geometry:

          ab=area of a rectangle of sides aand b

          abc= volume of a rectangular solid

          ___bh= area of a triangle with base band height h

         πr²h=volume of a cylinder with circular base of radius rand height h

         __πr²h=volume of cone with circular base of radius rand height h

         _____= distance from the origin to (x,y)

 

Physics:

          F=ma=force required to give a mass man acceleration a

          p(x,y,z)= density of a three- dimensional object at the point (x,y,z)

          F=Gm1m2/s²= gravitational force between objects of mass m1 and m2

                       at distance s

 

          m=______= relativistic mass of an object with rest mass m0and velocity v

 

Economics:

     c(x,y)= cost of xitems of one commodity and yitems of another commodity

     D1(p1, p2) = demand for commodity one when commodity one has price p1and commodity two has price p2

 

PROBLEMS  FOR  SECTION  1.2

For each of the following functions (Problems 1-8), make a table showing the value of f(x)when x= -1, ___ , 0, ___, 1. Put a * where f(x)is undefined. Example:

                           f(x)= ___                

 

 

1   f(x) =x/ 3                       2  f(x) = 3 

3  f(x) =3x3 - 5x2 +2                4   f(x) =1/(x -1) 

5    f(x) =____                    6   f(x) =|x| 

7    f(x) =|x - __| + |x+ __| 

8    f(x) =____

9    is the set of ordered pairs {3,2), (0,1), (4,2)} a function?

10   is the set of ordered pairs {0,2), (3,6), (3,4)} a function?

11   if f is the function f(x)=1+x+x², find f(2), f(t), f(t+Δt), f(1+t+t²), f(g(t)).

12   if f(x)=1/x, find f(t), f(t+Δt), f(t²), f(1/t), f(g(t)).

13   if f(x)=_______, find f(t), f(t+Δt), f(t²), f(____), f(g(t)).

14    if f(x)=ax+b, find f(ct+d), f(t²), f(1/t),f(t/a), f(g(t)).

For each of the following functions (Problems 15-20), find f(x+Δx) - f(x).

15    f(x)=4x+1  

16    f(x)=x²-x 

17    f(x)=x -2              18      f(x)=x4

19    f(x)=____          20    f(x)=4

21  find the domain of the function f(x) = 1/(x²-1)

22  find the domain of the function f(z) = ________.

23  what is the domain of the function f(x)= _____.

24  what is the domain of the function f(t)=______.

25  what is the domain of the function f(x)=______.

□26  show that if aand b have the same sign then |a+b|=|a|+|b|, and if a and b have opposite signs then |a+b|< |a|+|b|.