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Exponential and Logarithmic Functions. Chapter 4. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A. Composite Functions. Section 4.1. Composite Functions. Construct new function from two given functions f and g Composite function :

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## Exponential and Logarithmic Functions

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**Exponential and Logarithmic Functions**Chapter 4 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA**Composite Functions**Section 4.1**Composite Functions**• Construct new function from two given functions f and g • Compositefunction: • Denoted by f °g • Read as “f composed with g” • Defined by (f°g)(x) = f(g(x)) • Domain: The set of all numbers x in the domain of g such that g(x) is in the domain of f.**Composite Functions**• Note that we perform the inside function g(x) first.**Composite Functions**• Example. Suppose that f(x) = x3 { 2 and g(x) = 2x2 + 1. Find the values of the following expressions. (a) Problem: (f±g)(1) Answer: (b) Problem: (g±f)(1) Answer: (c) Problem: (f±f)(0) Answer:**Composite Functions**• Example. Suppose that f(x) = 2x2 + 3 and g(x) = 4x3 + 1. (a) Problem: Find f±g. Answer: (b) Problem: Find the domain of f±g. Answer: (c) Problem: Find g± f. Answer: (d) Problem: Find the domain of f±g. Answer:**Composite Functions**• Example. Suppose that f(x) = and g(x) = (a) Problem: Find f±g. Answer: (b) Problem: Find the domain of f±g. Answer: (c) Problem: Find g± f. Answer: (d) Problem: Find the domain of f±g. Answer:**Composite Functions**• Example. Problem: If f(x) = 4x + 2 and g(x) = show that for all x, (f±g)(x) = (g±f)(x) = x**Decomposing Composite Functions**• Example. Problem: Find functions f and g such that f±g = H if Answer:**Key Points**• Composite Functions • Decomposing Composite Functions**One-to-One Functions;Inverse Functions**Section 4.2**One-to-One Functions**• One-to-one function: Any two different inputs in the domain correspond to two different outputs in the range. • If x1 and x2 are two different inputs of a function f, then f(x1) f(x2).**One-to-One Functions**• One-to-one function • Not a one-to-one function • Not a function**One-to-One Functions**• Example. Problem: Is this function one-to-one? Answer: Person Salary Melissa John Jennifer Patrick $45,000 $40,000 $50,000**One-to-One Functions**• Example. Problem: Is this function one-to-one? Answer: Person ID Number 1451678 1672969 2004783 1914935 Alex Kim Dana Pat**One-to-One Functions**• Example. Determine whether the following functions are one-to-one. (a) Problem:f(x) = x2 + 2 Answer: (b) Problem:g(x) = x3 { 5 Answer:**One-to-One Functions**• Theorem. A function that is increasing on an interval I is a one-to-one function on I. A function that is decreasing on an interval I is a one-to-one function on I.**Horizontal-line Test**• If every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one.**Horizontal-line Test**• Example. Problem: Use the graph to determine whether the function is one-to-one. Answer:**Horizontal-line Test**• Example. Problem: Use the graph to determine whether the function is one-to-one. Answer:**Inverse Functions**• Requires f to be a one-to-one function • The inverse function of f • Written f{1 • Defined as the function which takes • f(x) as input • Returns the output x. • In other words, f{1 undoes the action of f • f{1(f(x)) = x for all x in the domain of f • f(f{1(x)) = x for all x in the domain of f{1**Inverse Functions**• Example. Find the inverse of the function shown. Problem: Person ID Number 1451678 1672969 2004783 1914935 Alex Kim Dana Pat**Inverse Functions**• Example. (cont.) Answer: ID Number Person 1451678 1672969 2004783 1914935 Alex Kim Dana Pat**Inverse Functions**• Example. Problem: Find the inverse of the function shown. f(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)g Answer:**Domain and Range of Inverse Functions**• If f is one-to-one, its inverse is a function. • The domain of f{1 is the range of f. • The range of f{1 is the domain of f**Domain and Range of Inverse Functions**• Example. Problem: Verify that the inverse of f(x) = 3x { 1 is**Graphs of Inverse Functions**• The graph of a function f and its inverse f{1 are symmetric with respect to the line y = x.**Graphs of Inverse Functions**• Example. Problem: Find the graph of the inverse function Answer:**Finding Inverse Functions**• If y = f(x), • Inverse if given implicitly by x = f(y). • Solve for y if possible to get y = f {1(x) • Process • Step 1: Interchange x and y to obtain an equation x = f(y) • Step 2: If possible, solve for y in terms of x. • Step 3: Check the result.**Finding Inverse Functions**• Example. Problem: Find the inverse of the function Answer:**Restricting the Domain**• If a function is not one-to-one, we can often restrict its domain so that the new function is one-to-one.**Restricting the Domain**• Example. Problem: Find the inverse of if the domain of f is x¸ 0. Answer:**Key Points**• One-to-One Functions • Horizontal-line Test • Inverse Functions • Domain and Range of Inverse Functions • Graphs of Inverse Functions • Finding Inverse Functions • Restricting the Domain**Exponential Functions**Section 4.3**Exponents**• For negative exponents: • For fractional exponents:**Exponents**• Example. Problem: Approximate 3¼ to five decimal places. Answer:**Laws of Exponents**• Theorem. [Laws of Exponents]If s, t, a and b are real numbers with a> 0 and b> 0, then • as¢at = as+t • (as)t = ast • (ab)s = as¢bs • 1s = 1 • a0 = 1**Exponential Functions**• Exponential function: function of the form f(x) = ax • where a is a positive real number (a> 0) • a 1. • Domain of f: Set of all real numbers. Warning! This is not the same as a power function. (A function of the form f(x) = xn)**Exponential Functions**• Theorem. For an exponential function f(x) = ax, a > 0, a 1, if x is any real number, then**Graphing Exponential Functions**• Example. Problem: Graph f(x) = 3x Answer:**Properties of the Exponential Function**• Properties of f(x) = ax, a > 1 • Domain: All real numbers • Range: Positive real numbers; (0, 1) • Intercepts: • No x-intercepts • y-intercept of y = 1 • x-axis is horizontal asymptote as x {1 • Increasing and one-to-one. • Smooth and continuous • Contains points (0,1), (1, a) and**Properties of the Exponential Function**f(x) = ax, a > 1**Properties of the Exponential Function**• Properties of f(x) = ax, 0 <a < 1 • Domain: All real numbers • Range: Positive real numbers; (0, 1) • Intercepts: • No x-intercepts • y-intercept of y = 1 • x-axis is horizontal asymptote as x 1 • Decreasing and one-to-one. • Smooth and continuous • Contains points (0,1), (1, a) and**Properties of the Exponential Function**f(x) = ax, 0 <a < 1**The Number e**• Number e: the number that the expression approaches as n1. • Use ex or exp(x) on your calculator.**The Number e**• Estimating value of e • n = 1: 2 • n = 2: 2.25 • n = 5: 2.488 32 • n = 10: 2.593 742 460 1 • n = 100: 2.704 813 829 42 • n = 1000: 2.716 923 932 24 • n = 1,000,000,000: 2.718 281 827 10 • n = 1,000,000,000,000: 2.718 281 828 46**Exponential Equations**• If au = av, then u = v • Another way of saying that the function f(x) = ax is one-to-one. • Examples. (a) Problem: Solve 23x {1 = 32 Answer: (b) Problem: Solve Answer:**Key Points**• Exponents • Laws of Exponents • Exponential Functions • Graphing Exponential Functions • Properties of the Exponential Function • The Number e • Exponential Equations

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