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1.5 Functions and Logarithms Golden Gate Bridge San Francisco, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington A relation is a function if: for each x there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one x. In other words, a function is one-to-one on domain D if: f a f b whenever a b To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 1 2 3 4 5 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 1 2 3 4 5 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 1 2 3 4 5 1 3 y x 2 1 2 y x 2 x y2 one-to-one not one-to-one not a function (also not one-to-one) Inverse functions: f x 1 x 1 2 Given an x value, we can find a y value. 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 1 2 3 4 5 1 y x 1 2 Solve for x: 1 y 1 x 2 2y 2 x x 2y 2 y 2x 2 Inverse functions are reflections about y = x. Switch x and y: f 1 x 2x 2 (eff inverse of x) example 3: Graph: f x x 2 x0 f x f 1 x yx for x0 a parametrically: Y= f : x1 t f 1 : x2 t 2 y x : x3 t y1 t 2 y2 t y3 t t0 WINDOW GRAPH example 3: Graph: f x x 2 x0 f x f 1 x yx for x0 b Find the inverse function: yx 2 x 0 WINDOW Switch x & y: yx x y y x f 1 x x y1 x2 x 0 Change the graphing mode to function. Y= y2 x y3 x > GRAPH Consider f x ax This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Example: 16 24 4 log 2 16 Two raised to what power is 16? The most commonly used bases for logs are 10: log10 x log x and e: log e x ln x y ln x is called the natural log function. y log x is called the common log function. In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work. y ln x is called the natural log function. y log x is called the common log function. Properties of Logarithms a log a x x log a a x x a 0 , a 1 , x 0 Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: log a xy log a x log a y x log a log a x log a y y log a x y log a x y Change of base formula: ln x log a x ln a Example 6: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? 1000 1.0525 2500 t 1.0525 t 2.5 ln 1.0525 ln 2.5 t We use logs when we have an unknown exponent. t ln 1.0525 ln 2.5 ln 2.5 t 17.9 ln 1.0525 17.9 years In real life you would have to wait 18 years. Example 7: Indonesian Oil Production (million barrels per year): 1960 20.56 1970 42.10 1990 70.10 Use the natural logarithm regression equation to estimate oil production in 1982 and 2000. How do we know that a logarithmic equation is appropriate? In real life, we would need more points or past experience. Indonesian Oil Production: 60 70 90 20.56 million 42.10 70.10 60,70,90 L1 2nd ENTER 2nd { 60,70,90 } STO alpha L 1 ENTER 20.56, 42.10,70.10 L2 LnReg L1, L2 ENTER 2nd MATH 6 3 5 alpha L 1 , alpha L 2 Done ENTER Statistics LnReg Regressions The calculator should return: ExpReg L1, L2 ENTER 2nd MATH 6 3 5 alpha L 1 , alpha L 2 Done ENTER Statistics LnReg Regressions The calculator should return: ShowStat ENTER 2nd MATH 6 8 ENTER Statistics ShowStat The calculator gives you an equation and constants: y a b ln x a 474.3 b 121.1 We can use the calculator to plot the new curve along with the original points: Y= 2nd y1=regeq(x) VAR-LINK x regeq ) Plot 1 ENTER ENTER WINDOW Plot 1 ENTER ENTER WINDOW GRAPH WINDOW GRAPH What does this equation predict for oil production in 1982 and 2000? F3 Trace This lets us see values for the distinct points. This lets us trace along the line. 82 ENTER Enters an x-value of 82. Moves to the line. In 1982, production was 59 million barrels. 100 ENTER Enters an x-value of 100. In 2000, production was 84 million barrels. p

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