chapter 2 - functions and graphs
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INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
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Chapter 2 Chapter 2 Functions and GraphsFunctions and Graphs
2007 Pearson Education Asia
INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
1. Applications and More Algebra
2. Functions and Graphs3. 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
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9. Additional Topics in Probability10. Limits and Continuity11. Differentiation12. Additional Differentiation Topics13. Curve Sketching14. Integration15. Methods and Applications of Integration16. Continuous Random Variables17. Multivariable Calculus
INTRODUCTORY MATHEMATICAL ANALYSIS
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• 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
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Functions
Special Functions
Combinations of Functions
Inverse Functions
Graphs in Rectangular Coordinates
Symmetry
Translations and Reflections
Chapter 2: Functions and Graphs
Chapter OutlineChapter Outline2.1)
2.2)
2.3)
2.4)
2.5)
2.6)
2.7)
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• 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
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Chapter 2: Functions and Graphs2.1 Functions
Example 1 – Determining Equality of Functions
Determine which of the following functions are equal.
1 if 3 1 if 2
)( d.
1 if 0 1 if 2
)( c.
2)( b.)1(
)1)(2()( a.
xxx
xk
xxx
xh
xxgx
xxxf
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Chapter 2: Functions and Graphs2.1 FunctionsExample 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
kfhfgf
khhgkg
, ,
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Chapter 2: Functions and Graphs2.1 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 5
g r r r r r
2
2 2( ) 3( ) ( ) 5
3 6 3 5
g x h x h x hx hx h x h
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Chapter 2: Functions and Graphs2.1 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: pq
q 100
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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
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Chapter 2: Functions and Graphs2.2 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 xf xx
( ) 2 3g x x 2 3 2 31
xx
Absolute-value function is defined as , e.g. xif 0if 0
x xx
x x
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Chapter 2: Functions and Graphs2.2 Special Functions
Example 7 – GeneticsTwo 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.
51 35!4 4( ) 0,1,2,...,5! 5 !
r r
P r rr r
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Chapter 2: Functions and Graphs2.2 Special FunctionsExample 7 – Genetic
Solution:3 21 3 1 95! 120
454 4 64 163!2! 6(2) 512
(3)P
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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 xf g x f x g xfg x f x g xf f xx g xg g x
If f(x) = 3x − 1 and g(x) = x2 + 3x, find a. ( )( ) b. ( )( ) c. ( )( )
d. ( )g1 e. ( )( )2
f g xf g xfg xf x
f x
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Chapter 2: Functions and Graphs2.3 Combinations of FunctionsExample 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 1d. ( )( ) 3
1 1 1 3 1e. ( )( ) ( ( )) (3 1)2 2 2
f g x f x g x x x x x xf g x f x g x x x x xfg x f x g x x x x x x xf f x xxg g x x x
xf x f x x
2
Composition
• Composite of f with g is defined by ( )( ) ( ( ))f g x f g x
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Chapter 2: Functions and Graphs2.3 Combinations of Functions
Example 3 – Composition
Solution:
2If ( ) 4 3, ( ) 2 1, and ( ) ,finda. ( ( ))b. ( ( ( )))c. ( (1))
F p p p G p p H p pF G pF G H pG 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 pF G p p p p p
G F G G
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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 axg f x g f x xa a
( )( ) ( ( )) ( )x bf g x f g x a b x b b xa
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Chapter 2: Functions and Graphs2.4 Inverse Functions
Example 3 – Inverses Used to Solve EquationsMany 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
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Chapter 2: Functions and Graphs2.4 Inverse Functions
Example 5 – Finding the Inverse of a FunctionTo 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.
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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.
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Chapter 2: Functions and Graphs2.5 Graphs in Rectangular Coordinates
Example 1 – Intercepts and GraphFind the x- and y-intercepts of the graph of y = 2x + 3, and sketch the graph.
Solution:When y = 0, we haveWhen x = 0,
30 2 3 so that 2
x x 2(0) 3 3y
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Chapter 2: Functions and Graphs2.5 Graphs in Rectangular Coordinates
Example 3 – Intercepts and GraphDetermine the intercepts of the graph of x = 3, and sketch the graph.
Solution:There is no y-intercept, because x cannot be 0.
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Chapter 2: Functions and Graphs2.5 Graphs in Rectangular Coordinates
Example 7 – Graph of a Case-Defined FunctionGraph the case-defined function
Solution:
if 0 < 3( ) 1 if 3 5
4 if 5 < 7
x xf x x x
x
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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.
2b a2 2( )a a b
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Chapter 2: Functions and Graphs2.6 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:
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Chapter 2: Functions and Graphs2.6 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.
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Chapter 2: Functions and Graphs2.6 SymmetryExample 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 41 01 or 1x
x x
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Chapter 2: Functions and Graphs2.6 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.
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Chapter 2: Functions and Graphs
2.7 Translations and Reflections2.7 Translations and Reflections• 6 frequently used functions:
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Chapter 2: Functions and Graphs2.7 Translations and Reflections
• Basic types of transformation:
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Chapter 2: Functions and Graphs2.7 Translations and Reflections
Example 1 – Horizontal TranslationSketch the graph of y = (x − 1)3.Solution: