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Barnett/Ziegler/Byleen Precalculus: A Graphing Approach

Barnett/Ziegler/Byleen Precalculus: A Graphing Approach. Chapter Four Inverse Functions: Exponential and Logarithmic Functions. Operations on Functions. The sum, difference, product, and quotient of the functions f and g are the functions defined by

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Barnett/Ziegler/Byleen Precalculus: A Graphing Approach

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  1. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Four Inverse Functions:Exponential and Logarithmic Functions

  2. Operations on Functions The sum,difference, product, and quotient of the functions f and g are the functions defined by (f + g)(x) = f(x) + g(x)Sum function (f – g)(x) = f(x) – g(x)Difference function (fg)(x) = f(x) g(x)Product function = Quotient function Each function is defined on the intersection of the domains of f and g, with the exception that the values of x where g(x) = 0 must be excluded from the domain of the quotient function. The composite of f and g is the function defined by (f  g) (x) = f [g(x)]Composite function The domain of f  g is the set of all real numbers x in the domain of g for which g(x) is in the domain of f. 4-1-31

  3. One-to-One Functions A function is one-to-one if no two ordered pairs in the function have the same second component and different first components. Horizontal Line Test A function is one-to-one if and only if each horizontal line intersects the graph of the function in at most one point. (a) f(a) = f(b) for ab; (b) Only one point has ordinate f is not one-to-one f(a); f is one-to-one 4-2-32

  4. Increasing and Decreasing Functions If a function f is increasing throughout its domain or decreasing throughout its domain, then f is a one-to-one function. (a) An increasing function is always one-to-one (c) A one-to-one function is not always increasing or decreasing (b) A decreasing function is always one-to-one 4-2-33

  5. Inverse of a Function If f is a one-to-one function, then the inverse of f, denoted f –1, is the function formed by reversing all the ordered pairs in f. Thus, f–1 = { (y, x) | (x, y) is in f } To find the inverse of a function f: Step 1. Find the domain of f and verify that f is one-to-one. If f is not one-to-one, then stop, since f –1 does not exist. Step 2. Solve the equation y = f(x) for x. The result is an equation of the form x = f –1(y). Step 3. Interchange x and y in the equation found in Step 2. This expresses f –1 as a function of x. Step 4. Find the domain of f –1. Remember, the domain of f –1 must be the same as the range of f. Check your work by verifying that f –1 [ f(x) ] = x for all x in the domain of f , and f [ f –1 (x)] = x for all x in the domain of f –1 4-2-34

  6. Exponential Graphs Basic Properties of the Graph of f(x) = bx, b > 0, b  1 1. All graphs will pass through the point (0, 1) since b0 = 1. 2. All graphs are continuous curves, with no holes or jumps. 3. The x axis is a horizontal asymptote. 4. If b > 1, then bxincreases as x increases. 5. If 0 < b < 1, then bx decreases as x increases. 6. The function f is one-to-one. 4-3-35

  7. x 1 æ ö ç 1 + ÷ x x è ø 1 2 10 2.59374… 100 2.70481… 1,000 2.71692… 10,000 2.71814… 100,000 2.71827… 1,000,000 2.71828… . . . . . . The Number e e = 2.718 281 828 459 π e 4-4-36

  8. The Exponential Function with Base e For x a real number, the equation f(x) = ex defines the exponential function with base e. The graphs of y = ex and y = e –x are shown in the figure. 4-4-37

  9. Exponential Growth and Decay Description Equation Graph Uses Short-term population growth (people, bacteria, etc.); growth of money at continuous compound interest Radioactive decay: light absorption in water, glass, etc.; atmospheric pressure; electric circuits y = cektc, k > 0 y = ce–ktc, k > 0 Unlimited growth Exponential decay 4-4-38(a)

  10. M = y –kt 1 + ce > 0 c, k, M Exponential Growth and Decay Description Equation Graph Uses y = c(1 – e–kt )c, k > 0 Learning skills; sales fads; company growth; electric circuits Long-term population growth; epidemics; sales of new products; company growth Limited growth Logistic growth 4-4-38(b)

  11. f x y = 2 y = x –1 f f x y x y = 2 x = 2 y 1 1 –3 –3 8 8 1 1 –2 –2 4 4 1 1 –1 –1 2 2 0 1 1 0 1 2 2 1 2 4 4 2 3 8 8 3 Ordered pairs reversed Logarithmic Function with Base 2 y 10 – 1 f y 5 x = 2 or y = log2x x –5 5 10 –5 –1 DOMAIN of f = (–, ) = RANGE of f –1 RANGE of f = (0, ) = DOMAIN of f 4-5-39

  12. Properties of Logarithmic Functions If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: 4-5-40

  13. The Decibel Scale The decibel level D of a sound of intensity I , measured in watts per square meter (W/ m2) is given by where I0 = 10–12 W/ m2 is the intensity of the least audible sound that an average healthy person can hear. Sound Intensity, W/ m2 Sound   1.0  10–12 Threshold of hearing 5.2  10–10 Whisper 3.2  10–6 Normal conversation 8.5  10–4 Heavy traffic 3.2  10–3 Jackhammer 1.0  100 Threshold of pain 8.3  102 Jet plane with afterburner 4-6-41

  14. The Richter Scale The magnitude M on the Richter scale of an earthquake that releases energy E , measured in joules, is given by where E0 = 104.40 joules is the energy released by a small reference earthquake. Magnitude on Richter scale Destructive power M < 4.5 Small 4.5 < M < 5.5 Moderate 5.5 < M < 6.5 Large 6.5 < M < 7.5 Major 7.5 < M Greatest 4-6-42

  15. Change-of-Base Formula 4-7-43

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