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Chapter 2 Functions and Graphs

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**INTRODUCTORY MATHEMATICAL ANALYSIS**

0. Review of Algebra Applications and More Algebra Functions and Graphs Lines, Parabolas, and Systems Exponential and Logarithmic Functions Mathematics of Finance Matrix Algebra Linear Programming Introduction to Probability and Statistics

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**INTRODUCTORY MATHEMATICAL ANALYSIS**

Additional Topics in Probability Limits and Continuity Differentiation Additional Differentiation Topics Curve Sketching Integration Methods and Applications of Integration Continuous Random Variables Multivariable Calculus

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**Chapter Objectives To understand what functions and domains are.**

Chapter 2: Functions and Graphs Chapter Objectives 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.

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**Chapter Outline Functions 2.1) Special Functions**

Chapter 2: Functions and Graphs Chapter Outline Functions Special Functions Combinations of Functions Inverse Functions Graphs in Rectangular Coordinates Symmetry Translations and Reflections 2.1) 2.2) 2.3) 2.4) 2.5) 2.6) 2.7)

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**Chapter 2: Functions and Graphs**

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): Domain of f = domain of g; f(x) = g(x).

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**Determine which of the following functions are equal.**

Chapter 2: Functions and Graphs 2.1 Functions Example 1 – Determining Equality of Functions Determine which of the following functions are equal.

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**By definition, g(x) = h(x) = k(x) for all x 1. **

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

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**Chapter 2: Functions and Graphs**

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). c. Find g(x + h).

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**Chapter 2: Functions and Graphs**

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:

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**2.2 Special Functions We begin with constant function.**

Chapter 2: Functions and Graphs 2.2 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.

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**b. is a rational function, since .**

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 Absolute-value function is defined as , e.g.

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**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.

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**Solution: Chapter 2: Functions and Graphs 2.2 Special Functions**

Example 7 – Genetic Solution:

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**2.3 Combinations of Functions**

Chapter 2: Functions and Graphs 2.3 Combinations of Functions Example 1 – Combining Functions We define the operations of function as: If f(x) = 3x − 1 and g(x) = x2 + 3x, find

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**Composite of f with g is defined by**

Chapter 2: Functions and Graphs 2.3 Combinations of Functions Example 1 – Combining Functions Solution: Composition Composite of f with g is defined by

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**Solution: Example 3 – Composition Chapter 2: Functions and Graphs**

2.3 Combinations of Functions Example 3 – Composition Solution:

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**2.4 Inverse Functions An inverse function is defined as**

Chapter 2: Functions and Graphs 2.4 Inverse Functions Example 1 – Inverses of Linear Functions An inverse function is defined as 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,

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**Applying f −1 to both sides gives . Since , is a solution.**

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.

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**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.

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**2.5 Graphs in Rectangular Coordinates**

Chapter 2: Functions and Graphs 2.5 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 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,

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**Determine the intercepts of the graph of x = 3, and sketch the graph.**

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.

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**Graph the case-defined function**

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:

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**Chapter 2: Functions and Graphs**

2.6 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. 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.

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**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:

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**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.

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**Solution: Replace y with –y, not equivalent to equation.**

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

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**A graph is symmetric about the y = x when (b, a) and (a, b).**

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.

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**2.7 Translations and Reflections**

Chapter 2: Functions and Graphs 2.7 Translations and Reflections 6 frequently used functions:

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**Basic types of transformation:**

Chapter 2: Functions and Graphs 2.7 Translations and Reflections Basic types of transformation:

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**Sketch the graph of y = (x − 1)3. Solution:**

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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