1. Mathematics for Business and Economics - I
Chapter 2 – Functions and Graphs

2. 2.1 Functions
The idea of a function is this: a correspondence between two sets D and R such that to each element of the first set, D, there corresponds one and only one element of the second set, R.
The first set is called the domain, and the set of corresponding elements in the second set is called the range.
Notation: if y is a function of x, we write
y = f(x)
Other common symbols for functions include but are not limited to g, h, F, G

3. Function Evaluation Consider our function
What does f (-3) mean? Replace x with the value –3 and evaluate the expression
The result is 7 . This means that the point (-3,7) is on the graph of the function.

4. Let g be the function defined by the equation y = x2 – 6x + 8.
EXAMPLE 1
Evaluating a Function
Let g be the function defined by the equation
y = x2 – 6x + 8.
Evaluate each function value.
Solution

5. EXAMPLE 1
Evaluating a Function
Solution continued

6. AGREEMENT ON DOMAIN
If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the largest set of real numbers that result in real numbers as outputs.

7. Domain of a Function Consider which is not a real number.
Question: for what values of x is the function defined?
Answer:
is defined only when the radicand (3x-2) is equal to or greater than zero. This implies that
Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3.

8. Example1: Find the domain of the function
Answer:
Example : Find the domain of
In this case, the function is defined for all values of x except where the denominator of the fraction is zero. This means all real numbers x except 5/3.

10. Find the domain of each function.
EXAMPLE 3
Finding the Domain of a Function
Find the domain of each function.
Solution
a. f is not defined when the denominator is 0.
Domain: {x|x ≠ –1 and x ≠ 1}

11. EXAMPLE 3
Finding the Domain of a Function
Solution continued
The square root of a negative number is not a real number and is excluded from the domain.
Domain: {x|x ≥ 0}, [0, ∞)
The square root of a negative number is not a real number and is excluded from the domain, so x – 1 ≥ 0. However, the denominator ≠ 0.

12. Domain: {x|x > 1}, or (1, ∞)
EXAMPLE 3
Finding the Domain of a Function
Solution continued
So x – 1 > 0 so x > 1.
Domain: {x|x > 1}, or (1, ∞)
Any real number substituted for t yields a unique real number.
Domain: {t|t is a real number}, or (–∞, ∞)

13. Find the domain of f(x) = x/(x2 –x – 2)
Example #2a (p.80)
Find the domain of f(x) = x/(x2 –x – 2)
The domain would be the set of all real numbers except those values of x which set the denominator equal to zero
These values are found by factoring
(x2 –x – 2) = (x + 1)(x - 2)
x = -1, 2
So the domain is the set of all real numbers , except x =-1, 2

14. Equality of Functions Two functions, f and g are equal (f = g) if
The domain of f is equal to the domain of g
For every x in the domain of f and g, the values of the two functions are the same; that is f(x) = g(x)

15. Example #1 (p. 79-80) Which of the following functions are equal
f(x) = (x + 2)(x + 1)/(x – 1)
g(x) = x + 2
h(x) = x + 2
Domains of g, h, the set of all real numbers and are equal,
but the domain of f is the set of all real numbers except x = 1

16. 1

17. 2 A polynomial function of degree n is a function of the form
where n is a nonnegative integer and the coefficients an, an–1, …, a2, a1, a0 are real numbers with a ≠ 0.

18. 3

19. 4

20. 2.3 Combinations of Functions
(i) Sum
(ii) Difference
(iii) Product
(iv) Quotient
(v)

21. EXAMPLE 1
Combining Functions

22. f(x) = x2 , g(x) = 3x, find; f(x) + g(x) = x2 + 3x f(x).g(x) = 3x3
EXAMPLE 2
Combining Functions
f(x) = x2 , g(x) = 3x, find;
f(x) + g(x) = x2 + 3x
f(x).g(x) = 3x3
f(x) – g(x) = x2 – 3x
f(x)/g(x) = x2/3x = x/3
cf(x) = cx2

23. Find each of the following functions.
EXAMPLE 3
Combining Functions
Let
Find each of the following functions.
Solution

24. EXAMPLE 3
Combining Functions
Solution continued

25. EXAMPLE 3
Combining Functions
Solution continued

26. COMPOSITION OF FUNCTIONS
If f and g are two functions, the composition of function f with function g is written as
and is defined by the equation
where the domain of
values x in the domain of g for which g(x) is in the domain of f.
consists of those

27. COMPOSITION OF FUNCTIONS

28. Find each of the following.
EXAMPLE 1
Evaluating a Composite Function
Let
Find each of the following.
Solution

29. EXAMPLE 1
Evaluating a Composite Function
Solution continued

30. Find each composite function.
EXAMPLE 2
Finding Composite Functions
Let
Find each composite function.
Solution

31. EXAMPLE 2
Finding Composite Functions
Solution continued

32. EXAMPLE 3
Finding the Domain of a Composite Function
Solution

33. Domain is (–∞, 0) U (0, ∞). Domain is (–∞, –1) U (–1, ∞).
EXAMPLE 3
Finding the Domain of a Composite Function
Solution continued
Domain is (–∞, 0) U (0, ∞).
Domain is (–∞, –1) U (–1, ∞).

34. following provides a decomposition of H(x).
EXAMPLE 4
Decomposing a Function
Show that each of the
following provides a decomposition of H(x).

35. EXAMPLE 4
Decomposing a Function
Solution

36. An ordered pair of real numbers is a pair of real numbers in which the order is specified, and is written by enclosing a pair of numbers in parentheses and separating them with a comma.
The ordered pair (a, b) has first component a and second component b. Two ordered pairs (x, y) and (a, b) are equal if and only if x = a and y = b.
The sets of ordered pairs of real numbers are identified with points on a plane called the coordinate plane or the Cartesian plane.

37. Definitions
We begin with two coordinate lines, one horizontal (x-axis) and one vertical (y-axis), that intersect at their zero points. The point of intersection of the x-axis and y-axis is called the origin. The x-axis and y-axis are called coordinate axes, and the plane formed by them is sometimes called the xy-plane.
The axes divide the plane into four regions called quadrants, which are numbered as shown in the next slide. The points on the axes themselves do not belong to any of the quadrants.

39. Definitions
The figure shows how each ordered pair (a, b) of real numbers is associated with a unique point in the plane P, and each point in the plane is associated with a unique ordered pair of real numbers. The first component, a, is called the x-coordinate of P and the second component, b, is called the y-coordinate of P, since we have called our horizontal axis the x-axis and our vertical axis the y-axis.

40. Definitions
The x-coordinate indicates the point’s distance to the right of, left of, or on the y-axis. Similarly, the y-coordinate of a point indicates its distance above, below, or on the x-axis. The signs of the x- and y-coordinates are shown in the figure for each quadrant. We refer to the point corresponding to the ordered pair (a, b) as the graph of the ordered pair (a, b) in the coordinate system. The notation P(a, b) designates the point P in the coordinate plane whose x-coordinate is a and whose y-coordinate is b.

41. Graph the following points in the xy-plane:
EXAMPLE 1
Graphing Points
Graph the following points in the xy-plane:
Solution
3 units right, 1 unit up
2 units left, 4 units up
3 units left, 4 units down
2 units right, 3 units down
3 units left, 0 units up or down

42. Solution continued EXAMPLE 1 Graphing Points
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

43. PROCEDURE FOR FINDING THE INTERCEPTS OF A GRAPH
Definitions
The points where a graph intersects (crosses or touches) the coordinate axes are of special interest in many problems. Since all points on the x-axis have a y-coordinate of 0, any point where a graph intersects the x-axis has the form (a, 0). The number a is called an x-intercept of the graph. Similarly, any point where a graph intersects the y-axis has the form (0, b), and the number b is called a y-intercept of the graph.
PROCEDURE FOR FINDING THE INTERCEPTS OF A GRAPH
Step1 To find the x-intercepts of an equation, set y = 0 in the equation and solve for x.
Step 2 To find the y-intercepts of an equation, set x = 0 in the equation and solve for y.

44. EXAMPLE 1
Finding Intercepts

45. Step 1 To find the x-intercepts, set y = 0, solve for x.
EXAMPLE 2
Finding Intercepts
Find the x- and y-intercepts of the graph of the equation y = x2 – x – 2.
Solution
Step 1 To find the x-intercepts, set y = 0, solve for x.
The x-intercepts are –1 and 2.

46. Step 2 To find the y-intercepts, set x = 0, solve for y.
EXAMPLE 2
Finding Intercepts
Solution continued
Step 2 To find the y-intercepts, set x = 0, solve for y.
The y-intercept is –2.

47. The graph of the linear equation(line)
The following steps can be used to draw the graph of a linear equation.
Step1 ) Select at least 2 values for x
Step2 ) Substitute them in the equation and find the corresponding values for y
Step3 ) Plot the points on cartesian plane
Step4 ) Draw a straight line through the points.

48. X -2 -1 1 y 3 5 5 3 1 -2 -1 1 -1

50. x-intercept
y=0  0=2x+3
x= 2/3
(2/3 , 0) is x-intercept
y- intercept
x=0  y=2.0+3
y= 3
(0,3) y-intercept
y-intercept
x-intercept

51. 3 -2

54. Step1

55. Step2
Step3
Step4
x-intercept
x-intercept
3/2 -2 5
y-intercept
Vertex point
-49/4

56. Step1
Step3
Step2

57. Step4
Step5

58. Ex : Sketch the Graph of

59. 2 4
Vertex point
y-intercept -8