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Chapter 4

Chapter 4. Exponential and Logarithmic Functions. 4.1. Inverse Functions. One-to-One Function In a one-to-one function, each x -value corresponds to only one y -value, and each y -value corresponds to only one x -value. Functions. 1 2 3. 4 5 6. One-to-one function. 2 4 6.

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Chapter 4

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  1. Chapter 4 Exponential and Logarithmic Functions

  2. 4.1 Inverse Functions

  3. One-to-One Function In a one-to-one function, each x-value corresponds to only one y-value, and each y-value corresponds to only one x-value. Functions 1 2 3 4 5 6 One-to-one function. 2 4 6 7 9 11 Not a one-to-one function.

  4. Example Determine whether a function defined by is one-to-one. Solution: If any y-value in a function corresponds to more than one x-value, then the function is not one-to-one. Here, a given y-value may correspond to more than one x-value. For example, both x = 2 and x = 2 corresponds to y = 4. Thus, by definition, f is not a one-to-one function.

  5. Horizontal Line Test Horizontal Line Test If any line intersects the graph of a function in no more than one point, then the function is one-to-one.

  6. Solution: Since more than one different values of x leads to the same value of y, the function is not one-to-one. Solution: Since every horizontal line will intersect the graph at exactly one point, this function is one-to-one. Example Determine whether the graphs are one-to-one functions.

  7. One-to-One Functions Test for One-to-One Functions • In a one-to-one function every y-value corresponds to no more than one x-value. To show that a function is not one-to-one, find at least two x-values that produce the same y-values. • Sketch the graph and use the horizontal line test. • If the function either increases or decreases on its entire domain, then it is one-to-one. A sketch is helpful here, too.

  8. Inverse Function Inverse Function Let f be one-to-one function. Then g is the inverse function of f if for every x in the domain of g, and for every x in the domain of f.

  9. Deciding Whether Two Functions Are Inverses Example: Let functions f and g be defined by f(x) = 3x + 9 and g(x) = , respectively. Is g the inverse function of f ? Solution:The horizontal line test applied to the graph indicates that f is one-to-one. Since it is one-to-one, we now find: Since function g is not the inverse of function f.

  10. Finding Inverses of One-to-One Functions Example: Find the inverse of the function if it is one-to-one. G = {(5,2), (3,1), (4,3), (2,0)} Solution: Every x-value in G corresponds to only one y-value., and every y-value corresponds to only one x-value, so G is a one-to-one function. The inverse function is found by interchanging the x- and y-values in each ordered pair. G1 ={(2,5), (1,3), (3,4), (0,2)} • Notice how the domain and range of G becomes the range and domain respectively, of G1 .

  11. Equations of Inverses Finding the Equation of the Inverse of y = f(x). For a one-to-one function f defined by an equation y = f(x), find the defining equation of the inverse as follows. Step 1 Interchange x and y, Step 2 Solve for y. Step 3 Replace y with f1 (x).

  12. f(x) = 3x + 7 Solution: The graph of y = 3x + 7 is a one-to-one function, verified by the horizontal line test. y = 3x + 7 x = 3y + 7 3y = x 7 y = f1(x) = y = x2 5 Solution: The equation y = x2  5 has a parabola opening down as its graph, so some horizontal lines will intersect the graph at two points. This function is not one-to-one and will not have an inverse. Finding Equations of Inverses

  13. Finding the Inverse of a Function with a Restricted Domain Example: Let f(x) = Find f–1 (x). Solution: First, notice that the domain of f is restricted to the interval [3, ). Function f is one-to-one because it is increasing on its entire domain, and thus has an inverse function.

  14. Finding the Inverse of a Function with a Restricted Domain continued • However, we cannot define f 1(x) as x2 3. The domain of f is [3, ), and its range is [0, ). The range of f is the domain of f 1, so f 1 must be defined as f 1(x) = x2  3, x  0.

  15. Inverses Important Facts about Inverses • If f is one-to-one, then f 1 exists. • The domain of f is equal to the range of f 1, andthe range of f is equal to the domain of f 1. • If the point (a, b) lies on the graph of f, then (b, a) lies on the graph of f 1,, so the graph of f and f 1 are reflections of each other across the line y = x. • To find the equation for f 1, replace f(x) with y, interchange x and y, and solve for y. This gives f 1(x).

  16. 4.2 Exponential Functions

  17. Additional Properties of Exponents • If any real number a > 0, a 1, the following statements are true. • ax is a unique real number for all real numbers x. • ab = acif and only if b = c. • If a > 1 and m < n, then am < an. • If 0 < a < 1 and m < n, then am > an.

  18. Evaluating an Exponential Expression • If f(x) = 3x, find each of the following. • a) f(1) b) f(3) c) f(3/2) d) f(5.01) • Solutions: • a) b) • c) d)

  19. Exponential Function

  20. Exponential Function continued

  21. Characteristics of the Graph of f(x) = ax • 1. The points (0, 1), and (1, a) are on the graph. • 2. If a > 1, then f is an increasing function; if 0 < a < 1, then f is a decreasing function. • 3. The x-axis is a horizontal asymptote. • 4. The domain is (, ), and the range is (0, ).

  22. Graph f(x) = 6x. y-intercept = 1 x-axis = horizontal asymptote Domain: (, ) Range: (0, ) x f(x) 1 0.16 0 1 0.5 2.45 1 6 Example

  23. Graphing Reflections and Translations • Graph each function. • a) f(x) = 3x • b) f(x) = 3x + 2 • c) f(x) = 3x + 2

  24. f(x) = 3x Reflected across the x-axis. Domain: (, ) Range: (, 0) Solution

  25. f(x) = 3x + 2 The graph of f(x) = 3xtranslated 2 units to the left. Solution

  26. f(x) = 3x + 2 The graph of f(x) = 3xtranslated 2 units up. Solution

  27. Exponential Equations • Solve

  28. Another Example • Solve3x + 1 = 27x  3

  29. Compound Interest • If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) m times per year, then after t years the account will contain A dollars, where

  30. Example • Suppose $2000 is deposited in an account paying 6% compounded semiannually (twice a year). • a) Find the amount in the account after 10 years. • b) How much interest is earned over the 10-yr period?

  31. Solution a) • Thus, $3612.22 is an account after 10 yr. b) The interest earned for that period is $3612.22  $2000 = $1612.22

  32. Definitions • Value of e: To nine decimal places, e 2.718281828 • Continuous Compounding If P dollars are deposited at a rate of interest r compounded continuously for t years, the compounded amount in dollars on deposit is A = Pert.

  33. Example • Suppose $2000 is deposited in an account paying 2% interest compounded continuously for 3 yr. Find the total amount on deposit at the end of the 3 yr. • Solution:

  34. 4.3 Logarithmic Functions

  35. Logarithm • For all real numbers y, and all positive numbers a and x, where a 1: • Meaning of logax A logarithm is an exponent; logax is the exponent to which the base a must be raised to obtain x.

  36. Solving Logarithmic Equations • Solve each equation. • a) b)

  37. Logarithmic Function • If a > 0, a 1, and x > 0, then defines the logarithmic function with base a. Logarithms can be found for positive numbers only.

  38. Logarithmic Function

  39. Logarithmic Function continued

  40. Characteristics of the Graph of f(x) = logax • The points (1, 0), and (a, 1) are on the graph. • If a > 1, then f is an increasing function; if 0 < a < 1, then f is a decreasing function. • The y-axis is a vertical asymptote. • The domain is (0, ), and the range is (, ).

  41. Graph Write in exponential form as Now find some ordered pairs. x y 1 0 1/16 2 4 1 Example

  42. Graph Write in exponential form as Now find some ordered pairs. x y 1 0 5 1 0.2 1 Example

  43. Graph the function. The vertical asymptote is x = 1. To find some ordered pairs, use the equivalent exponent form. Translated Logarithmic Functions

  44. Graph To find some ordered pairs, use the equivalent exponent form. Translated Logarithmic Functions continued

  45. Property Description Product Property The logarithm of a product of two numbers is equal to the sum of the logarithms of the numbers Quotient Property The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers. Power Property The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. Properties of Logarithms, For x > 0, y > 0, a > 0, a 1, and any real number r:

  46. Using the Properties of Logarithms • Rewrite each expression. Assume all variables represent positive real numbers with a 1 and b  1. • a) • b) • c)

  47. Using the Properties of Logarithms • Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with a 1 and b  1. • a) • b)

  48. Theorem on Inverses • For a > 0, a 1: • By the results of this theorem:

  49. 4.4 Evaluating Logarithms and the Change-of-Base Theorem

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