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- 1. INTRODUCTORY MATHEMATICALINTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences ©2007 Pearson Education Asia Chapter 2Chapter 2 Functions and GraphsFunctions and Graphs
- 2. ©2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of Algebra 1. Applications and More Algebra 2. Functions and Graphs 3. Lines, Parabolas, and Systems 4. Exponential and Logarithmic Functions 5. Mathematics of Finance 6. Matrix Algebra 7. Linear Programming 8. Introduction to Probability and Statistics
- 3. ©2007 Pearson Education Asia 9. Additional Topics in Probability 10. Limits and Continuity 11. Differentiation 12. Additional Differentiation Topics 13. Curve Sketching 14. Integration 15. Methods and Applications of Integration 16. Continuous Random Variables 17. Multivariable Calculus INTRODUCTORY MATHEMATICAL ANALYSIS
- 4. ©2007 Pearson Education Asia • To understand what functions and domains are. • To introduce different types of functions. • To introduce addition, subtraction, multiplication, division, and multiplication by a constant. • To introduce inverse functions and properties. • To graph equations and functions. • To study symmetry about the x- and y-axis. • To be familiar with shapes of the graphs of six basic functions. Chapter 2: Functions and Graphs Chapter ObjectivesChapter Objectives
- 5. ©2007 Pearson Education Asia Functions Special Functions Combinations of Functions Inverse Functions Graphs in Rectangular Coordinates Symmetry Translations and Reflections Chapter 2: Functions and Graphs Chapter OutlineChapter Outline 2.1) 2.2) 2.3) 2.4) 2.5) 2.6) 2.7)
- 6. ©2007 Pearson Education Asia • A function assigns each input number to one output number. • The set of all input numbers is the domain of the function. • The set of all output numbers is the range. Equality of Functions • Two functions f and g are equal (f = g): 1.Domain of f = domain of g; 2.f(x) = g(x). Chapter 2: Functions and Graphs 2.1 Functions2.1 Functions
- 7. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.1 Functions Example 1 – Determining Equality of Functions Determine which of the following functions are equal. = ≠+ = = ≠+ = += − −+ = 1if3 1if2 )(d. 1if0 1if2 )(c. 2)(b. )1( )1)(2( )(a. x xx xk x xx xh xxg x xx xf
- 8. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.1 Functions Example 1 – Determining Equality of Functions Solution: When x = 1, By definition, g(x) = h(x) = k(x) for all x ≠ 1. Since g(1) = 3, h(1) = 0 and k(1) = 3, we conclude that ( ) ( ) ( ) ( ) ( ) ( )11 ,11 ,11 kf hf gf ≠ ≠ ≠ kh hg kg ≠ ≠ = , ,
- 9. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.1 Functions Example 3 – Finding Domain and Function Values Let . Any real number can be used for x, so the domain of g is all real numbers. a. Find g(z). Solution: b. Find g(r2 ). Solution: c. Find g(x + h). Solution: 2 ( ) 3 5g x x x= − + 2 ( ) 3 5g z z z= − + 2 2 2 2 4 2 ( ) 3( ) 5 3 5g r r r r r= − + = − + 2 2 2 ( ) 3( ) ( ) 5 3 6 3 5 g x h x h x h x hx h x h + = + − + + = + + − − +
- 10. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.1 Functions Example 5 – Demand Function Suppose that the equation p = 100/q describes the relationship between the price per unit p of a certain product and the number of units q of the product that consumers will buy (that is, demand) per week at the stated price. Write the demand function. Solution: p q q = 100
- 11. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.2 Special Functions2.2 Special Functions Example 1 – Constant Function • We begin with constant function. Let h(x) = 2. The domain of h is all real numbers. A function of the form h(x) = c, where c = constant, is a constant function. (10) 2 ( 387) 2 ( 3) 2h h h x= − = + =
- 12. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.2 Special Functions Example 3 – Rational Functions Example 5 – Absolute-Value Function a. is a rational function, since the numerator and denominator are both polynomials. b. is a rational function, since . 2 6 ( ) 5 x x f x x − = + ( ) 2 3g x x= + 2 3 2 3 1 x x + + = Absolute-value function is defined as , e.g.x if 0 if 0 x x x x x ≤ = − <
- 13. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.2 Special Functions Example 7 – Genetics Two black pigs are bred and produce exactly five offspring. It can be shown that the probability P that exactly r of the offspring will be brown and the others black is a function of r , On the right side, P represents the function rule. On the left side, P represents the dependent variable. The domain of P is all integers from 0 to 5, inclusive. Find the probability that exactly three guinea pigs will be brown. ( ) 5 1 3 5! 4 4 ( ) 0,1,2,...,5 ! 5 ! r r P r r r r − ÷ ÷ = = −
- 14. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.2 Special Functions Example 7 – Genetic Solution: 3 2 1 3 1 9 5! 120 454 4 64 16 3!2! 6(2) 512 (3)P ÷ ÷ ÷ ÷ = ==
- 15. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.3 Combinations of Functions2.3 Combinations of Functions Example 1 – Combining Functions • We define the operations of function as: ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ). ( ) ( ) ( ) for ( ) 0 ( ) f g x f x g x f g x f x g x fg x f x g x f f x x g x g g x + = + − = − = = ≠ If f(x) = 3x − 1 and g(x) = x2 + 3x, find a. ( )( ) b. ( )( ) c. ( )( ) d. ( ) g 1 e. ( )( ) 2 f g x f g x fg x f x f x + −
- 16. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.3 Combinations of Functions Example 1 – Combining Functions Solution: 2 2 2 2 2 3 2 2 a. ( )( ) ( ) ( ) (3 1) ( +3 ) 6 1 b. ( )( ) ( ) ( ) (3 1) ( +3 ) 1 c. ( )( ) ( ) ( ) (3 1)( 3 ) 3 8 3 ( ) 3 1 d. ( ) ( ) 3 1 1 1 3 1 e. ( )( ) ( ( )) (3 1) 2 2 2 f g x f x g x x x x x x f g x f x g x x x x x fg x f x g x x x x x x x f f x x x g g x x x x f x f x x + = + = − + = + − − = − = − − = − − = = − + = + − − = = + − = = − = 2 Composition • Composite of f with g is defined by ( )( ) ( ( ))f g x f g x=o
- 17. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.3 Combinations of Functions Example 3 – Composition Solution: 2 If ( ) 4 3, ( ) 2 1, and ( ) ,find a. ( ( )) b. ( ( ( ))) c. ( (1)) F p p p G p p H p p F G p F G H p G F = + − = + = 2 2 2 2 2 a. ( ( )) (2 1) (2 1) 4(2 1) 3 4 12 2 ( )( ) b. ( ( ( ))) ( ( ))( ) (( ) )( ) ( )( ( )) ( )( ) 4 12 2 4 12 2 c. ( (1)) (1 4 1 3) (2) 2 2 1 5 F G p F p p p p p F G p F G H p F G H p F G H p F G H p F G p p p p p G F G G = + = + + + − = + + = = = = = = + + = + = = + × − = = × + = o o o o o o o
- 18. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.4 Inverse Functions2.4 Inverse Functions Example 1 – Inverses of Linear Functions • An inverse function is defined as 1 1 ( ( )) ( ( ))f f x x f f x− − = = Show that a linear function is one-to-one. Find the inverse of f(x) = ax + b and show that it is also linear. Solution: Assume that f(u) = f(v), thus . We can prove the relationship, au b av b+ = + ( ) ( )( ) ( ( )) ax b b ax g f x g f x x a a + − = = = =o ( )( ) ( ( )) ( ) x b f g x f g x a b x b b x a − = = + = − + =o
- 19. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.4 Inverse Functions Example 3 – Inverses Used to Solve Equations Many equations take the form f(x) = 0, where f is a function. If f is a one-to-one function, then the equation has x = f −1(0) as its unique solution. Solution: Applying f −1 to both sides gives . Since , is a solution. ( )( ) ( )1 1 0f f x f− − = 1 (0)f −1 ( (0)) 0f f − =
- 20. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.4 Inverse Functions Example 5 – Finding the Inverse of a Function To find the inverse of a one-to-one function f , solve the equation y = f(x) for x in terms of y obtaining x = g(y). Then f−1 (x)=g(x). To illustrate, find f−1 (x) if f(x)=(x − 1)2 , for x ≥ 1. Solution: Let y = (x − 1)2 , for x ≥ 1. Then x − 1 = √y and hence x = √y + 1. It follows that f−1 (x) = √x + 1.
- 21. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.5 Graphs in Rectangular Coordinates2.5 Graphs in Rectangular Coordinates • The rectangular coordinate system provides a geometric way to graph equations in two variables. • An x-intercept is a point where the graph intersects the x-axis. Y-intercept is vice versa.
- 22. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.5 Graphs in Rectangular Coordinates Example 1 – Intercepts and Graph Find the x- and y-intercepts of the graph of y = 2x + 3, and sketch the graph. Solution: When y = 0, we have When x = 0, 3 0 2 3 so that 2 x x= + = − 2(0) 3 3y = + =
- 23. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.5 Graphs in Rectangular Coordinates Example 3 – Intercepts and Graph Determine the intercepts of the graph of x = 3, and sketch the graph. Solution: There is no y-intercept, because x cannot be 0.
- 24. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.5 Graphs in Rectangular Coordinates Example 7 – Graph of a Case-Defined Function Graph the case-defined function Solution: if 0 < 3 ( ) 1 if 3 5 4 if 5 < 7 x x f x x x x ≤ = − ≤ ≤ ≤
- 25. ©2007 Pearson Education Asia Use the preceding definition to show that the graph of y = x2 is symmetric about the y-axis. Solution: When (a, b) is any point on the graph, . When (-a, b) is any point on the graph, . The graph is symmetric about the y-axis. Chapter 2: Functions and Graphs 2.6 Symmetry2.6 Symmetry Example 1 – y-Axis Symmetry • A graph is symmetric about the y-axis when (-a, b) lies on the graph when (a, b) does. 2 b a= 2 2 ( )a a b− = =
- 26. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.6 Symmetry • Graph is symmetric about the x-axis when (x, -y) lies on the graph when (x, y) does. • Graph is symmetric about the origin when (−x,−y) lies on the graph when (x, y) does. • Summary:
- 27. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.6 Symmetry Example 3 – Graphing with Intercepts and Symmetry Test y = f (x) = 1− x4 for symmetry about the x-axis, the y-axis, and the origin. Then find the intercepts and sketch the graph.
- 28. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.6 Symmetry Example 3 – Graphing with Intercepts and Symmetry Solution: Replace y with –y, not equivalent to equation. Replace x with –x, equivalent to equation. Replace x with –x and y with –y, not equivalent to equation. Thus, it is only symmetric about the y-axis. Intercept at 4 1 0 1 or 1 x x x − = = = −
- 29. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.6 Symmetry Example 5 – Symmetry about the Line y = x • A graph is symmetric about the y = x when (b, a) and (a, b). Show that x2 + y2 = 1 is symmetric about the line y = x. Solution: Interchanging the roles of x and y produces y2 + x2 = 1 (equivalent to x2 + y2 = 1). It is symmetric about y = x.
- 30. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.7 Translations and Reflections2.7 Translations and Reflections • 6 frequently used functions:
- 31. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.7 Translations and Reflections • Basic types of transformation:
- 32. ©2007 Pearson Education Asia Chapter 2: Functions and Graphs 2.7 Translations and Reflections Example 1 – Horizontal Translation Sketch the graph of y = (x − 1)3 . Solution:

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