Bussiness Mathematics

Function and graphs

2.1 Function

·         A function is a rule that assign to each input number exactly one output number.

·         The set of all input numbers to which the rule applies is called domain of the function.

·         The set of all input numbers is called the range.

·         A variable that represents input numbers for a function is called an independent variable.

·         A variable that represent output numbers is called a dependent variable.

As example :

y = x + 5

defines y as a function of x. The quation gives the rule, “add 5 to x”. This rule assigns to each input x exactly one output x+5, which is y.  If x = 1, then y = 6 . if x = -3 then y = 2. The independent variable is x and the dependent variable is y.

Commonly, we usually write the function with notation : f(x) , means the output number in the range of f that corresponds to the input number x in domain. Thus the output f(x) is the same as y. But since y = x + 5, we can write y`= f(x) = x + 5 more simply .

f(x) = x+5

for example, to find f(3), which is the output corresponding to the input 3, we replace each x in

 f(x) = x+5 by 3.

f(3) = 3 + 5

f(3) = 8

output numbers such as f(-4) are called function value.

Equality of functions :

To say that two functions f and g are equal, denoted f = g, is to say that :

1.   The domain of f is equal to the domain of g.

2.  For every x in the domain of f  and  g, f(x) = g(x)

Problems 2.1

2.2  Special Functions

      In this section, we look at functions having special forms and representations. We begin with perhaps the simplest type of function there is: a constant function.

·         Example 1 Constant Functions

Let h(x) = 2. The domain of h is all real numbers. All function values are 2. For example,

h(10) = 2        h(-387) = 2     h(x+3) = 2

We call h a constant function because all the function values are the same. More generally, we have this definition:

A function of the form h(x) = c, where c is a constant, is called a constant function

     A constant function belongs to a broader class of functions, called polynomial functions. In general, a function of the form

f(x) = c_nxⁿ + c_n-1x^n-1 + … + c₁x + c₀

where n is a nonnegative integer and c_n,c_n-1,…,c₀ are constants with c_{n ≠} 0, is called a polynomial function (in x). the number n is called the degree of the polynomial, and c_n is the leading coefficient. Thus,

f(x) = 3x² - 8x + 9

is a polynomial function of degree 2 with leading coefficient 3. Likewise, g(x) = 4 - 2x has degree 1 and leading coefficient -2. Polynomial functions of degree 1 or 2 are called linear or quadratic functions, respectively.

Problem 2.2

2.3 combination of function

There are saveral ways of combining two functions to create a new function. Suppose f and g are the function given by

f(x) = x² and g(x) = 3x

adding f(x) and g(x)  gives

                        f(x) + g(x) = x² + 3x

this operation defines a new function called the sum af f and g, denoted

f + g. Its function value at x is f(x) + g(x). That is,

(f + g)(x) = f(x) + g(x) = X² + 3x

In general, for any function f and g, we define the sum f + g, the difference f – g, the product fg, and the quotient  as follows.

(f + g)(x) = f(x) + g(x)

(f – g)(x) = f(x) – g(x)

(fg)(x) = f(x).g(x)

 =  for g(x) ≠ 0

Example :

If f(x) = 3x – 1 and g(x) = x² + 3x, find (f + g)(x) !

(f + g)(x) = f(x) + g(x)

            = (3x – 1) + (x² + 3x)

            = x² + 6x – 1

Problem 2.3     

2.4   Inverse Function

Just as –a is the number for which

a + (-a) = 0 = (-a) + a

and, for a ≠ 0, a^-1 is the number for which

aa^-1 = 1 = a^-1a

so, given a function f, we can inquire about the existence of a function g satisfying

f ° g = I = g ° f

where I is the identity function, introduced in the subsection titled “Composition” of section 2.3 and given by I(x) = x. Suppose that we have g as above and a function h that also satisfies the equations of (1) so that

f ° h = I = h ° f

then

h = h ° I = h ° (f ° g) = (h ° f) ° g = I ° g = g

shows that there is at most one function satisfying the requirement of g in (1). In mathematical jargon, g is uniquely determined by f and is therefore given a name. g = f^—1, that reflects ^its dependence on f. The function f^—1 is read as f inverse and called the inverse of f.

A function of f that satisfies

For all a and b, if f(a) = f(b) then a = b

Is called a one-to-one function.

Problem 2.4

2.5 Graphs in Rectagular Coordinates

       A rectangular coordinate system allows us to specify and locate points in a plane. In a plane, two real-number lines, called coordinate axes, are costructed perpendicular to each other so that their origins coincide, as in Figure 2.7. Their point of intersection is called the origin of the coordinate system. For now, we will call the horizontal line the x-axis and the vertical line the y-axis.  The plane on which the coordinate axes are placed is called a rectangular coordinate plae or, more simply, an x,y-plane. To label point P in Figure 2.8(a), we draw perpendiculars from P to the x-axis and y-axis. They meet these axes at 4 and 2, respectively. Thus, P determines two number, 4 and 2. We say that the rectangular coordinates of P are given by the ordered pair (4,2) and Figure 2.8(b), the point corresponding to (4.2) is not the same as that for (2,4)

(4,2) ≠ (2,4)

In Figure 2.10, the coordinates of various points are indicated. For example, the point (1,-4) is located one unit to the right of the y-axis and four units below the x-axis. The coordinate axes divide the plane into four regions called quadrants (Figure 2.11). for example, quadrant I consists of all points (x₁,y₁) with x₁  > 0 and y₁ > 0. The points on the axes do not lie in any quadrant. Using a rectangular coordinate system, we can geometrically represent equations in two variables. For example, let us cosider y = x² + 2x – 3....(1). A solution of this equation is a value of x and a value of y that make the equation true. For example, if x = 1, substituting in to Equation (1) gives y = 1² + 2(1) – 3 = 0.

Thus, x = 1, y = 0 is a solution of Equation (1). Similarly, If x = -2 then y = (-2)²  + 2(-2) – 3 = -3. And so x = -2, y = -3 is also a solution. By choosing other values for x, we can get more solutions. [See Figure 2.12 (a).] it should be clear that there are infinitely many solutions of Equation (1). Each solution gives rise to a point (x,y). For example, to x = 1 and y = 0 corresponds (1,0). The graph of y = x² + 2x – 3 is the geometric representation of all its solutions. In Figure 2.12(b), we have plotted the points corresponding to the points corresponding to the solutions in the table. Then we join these points by a smooth curve wherever conditions permit. This gives the curve in figure 2.12(c). The point (0,-3) where the curve interects the y-axis is called the y-intercept. The points (-3,0) and (1,0) where the curve intersects the x-axis are called the x-intercepts. In general, we have the following definition.

	DEFINITION An x-intercept of the graph of an equation in x and y is a point where the graph intersects the x-axis. A y-intercept is a point where the graph intersects the y-axis.

 

The find the x-intercepts of the graph of an equation in x and y, we first set y = 0 and solve the resulting equation for x. To find the y –intercepts, we first set x = 0 and solve for y. For example, let us find the x-intercepts for the graph of y = x² + 2x – 3. Setting y = 0 and solving for x gives

0 = x² + 2x – 3

0 = (x+3) (x-1)

X = -3 , 1

Thus, the x-intercepts are (-3,0) and (1,0), as we saw before. If x = 0, then y = 0² + 2(0) – 3 = -3. So (0,-3) is the y-intercept. Keep in mind that an x-intercepts has its y-coordinate 0, and a y-intercepts has its x-coordinate 0. Intercepts are useful because they indicate prcisely where the graph intersects the axes.

·         EXAMPLE for Intercepts and Graph

Find the x- and y-intercepts of the graph of y = 2x + 3, and sketch the graphs.

Solution: If y = 0, then

0 = 2x + 3 so that x = - 3/2

Thus, the x-intercept is (-3/2,0). If x = 0, then

Y = 2(0) + 3 =3

So, the y-intercept is (0,3). Figure 2.13 shows a table of some points on the graph and a sketch of the graph.

·         EXAMPLE for Graph of the Square-Root Function

Graph f(x) = Öx

Solutin: The graph is shown in Figure 2.16. We label the vertical axis as f (x). Recall that Öx denotes the principal square root of x. Thus, f(9) = Ö9 = 3. Also, wd can’t choose negative values for x, because we don’t want imaginary numbers for Öx. That is, we must have x > 0. Let us now consider intercepts. If f (x) = 0, then Öx = 0, or x = 0. Also, if x = 0, then f (x) = 0. Thus, the x-intercept the vertical-axis intercept are the same, namely, (0,0).

·         EXAMPLE for Graph of the Absolute-value Function

Graph p = G (q) = lql.

Solution: We use the indepedent variable q to label the horizontal axis. The function-value axis can be labeled either G (q) or p. (See Figure 2.17). Notice that the q- and p-intercepts are the same point, (0,0).

For example, a zero of the function f (x) = 2x – 6 is 3 because f (3) = 2(3) – 6 = 0. Here we call 3 a real zero because it’s a real number. We note that zeros of f can be foundby setting f(x) = 0 and solving for x. Thus, the zeros of a function are precisely the x-intercepts of its graph, because it’s at these points that f (x) = 0. To further illustrate, Figure 2.18 shows the graph of the funtion y = f (x) = x² – 2x – 3. The x-intercepts of the graph are -1 and 3. Hence, -1 and 3 are zeros of f, or equivalently, -1 and 3 are solution to the equation x² – 2x – 3 = 0.

·         EXAMPLE for Graph of a Case-Defined Function

Graph the case-defined function

                           X if 0 < x < 3

              F (x) =          X – 1 if 3 < x < 5

                           4 if 3 < x < 5

Solution: The domain of f is 0 < x < 7. The graph is given in Figure 2.23, where the hollow dot means that the point is not icluded in the graph. Notice that the range of f is all real numbers y such that 0 < y < 4.

In Figure 2.24 (a), notice that with the given x there are associated two values of y:y₁ and y₂. Thus, the curve is not the graph of function of x. Looking at it another way, we have the following general rule, called the vertical-line test. If a vertical line L can be drawn that intersects a curve in at least two point, then the curve is not the graph of a function of x. When no such vertical lie can be drawn, the curve is the graph of a fuction of x. Consenquently, the curves in Figure 2.24 do not represent function of x, but those in figure 2.25 do.

EXAMPLE 8: A Graph That Does Not Represent a Function of x

Graph: x = 2y².

Solution: Here it is easier to chose values of y and then find the corresponding values of x. Figure 2.26 shows the graph. By the vertical-line test, the equation x = 2y² does not define a fuction of x. If f (4) = 3 and, apparently, also f (-4) = 3. Since the distinct input values -4 and 4 produce the same output, the function is not one-to-one. Looking at it another way, we have the following geeral rule, called the horizontal-line test. If a horizontal line L can be drawn that intersects the graph of a function in at least two points, then the function isn’t one-to-one. When no scuh horizontal line can be drawn, the function is one-to-one.

PROBLEM 2.5

2.6 Symmetry

In this basic part of mathematics "symmetry", we will see that calculus is a great aid in graphing
not only because it helps determine the shape of a graph, but also provides powerful technique for determining whether or not a curve "wiggles" between point.
this is some example that related to symmetry graph

·         y-axis symmetry

consider the graph of y = x² 
the portion to the left of the y-axis  is the reflection through the y-axis 
of that portion to the right of the y-axis .

so a graph is symmetric about the y-axis if and only if (-x,y) lies on the graph when (a,b) does

·         x-axis symmetry

 in the other hand, the x-axis just like the y-axis 
but the changes is in the y point 
so a graph is symmetric about the x-axis if and only if (x,-y) lies on the graph when (x,y) does.

·         Origin symmetry

this type maybe rather difficult , but actually its quite simply
for example from the picture above,
the graph whenever the point (x,y) lies on the graph,(-x,-y)also lies on it

2.7 Translations and Reflections

now we gonna learn another technique .
but this technique is not necessarily the preferred way. However ,some functions and their asociated graph occur so frequently that we find it worthwhile to learn it


to make it simply , this is how it works

equation
 y = f(x) + c   (shift c unit upward)
y = f(x) - c     (shift c unit downward)

y = f(x - c)      ( shift c unit to the right )
y = f(x + c )    (shift c unit to the left )

y = -f(x)       (reflect about x-axis)
y = f(-x)       (reflect about y-axis)

y = c f (x)  c>1 (vertically stretch away from x-axis by a factor of c )
y = c f (x)  c<1 (vertically shrink toward x-axis by a factor of c)

·         Horizontal Translation

take this example from the graph
the blue line f(x+2) is on the left to make it move to the right we can use equation y = f(x-c)
so the result is the green line f(x). this techniques used the same way if you want to move the graph to the right . if you do , that makes the graph to be f(x-2) 
the c unit to make ow far you want to move the graph
so if you want to make a far away little bit, you used the larger number and if you want to move it to the left
just simply lower the c  unit number