H-SC MATH 262 - Lecture 38 Notes - Real-Valued Functions

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Real-Valued Functions of a Real Variable and Their GraphsFunctionsPower FunctionsThe Constant Function f(x) = 1The Linear Function f(x) = xThe Quadratic Function f(x) = x2The Cubic Function f(x) = x3Power Functions xa, a  1The Square-Root FunctionThe Cube-Root FunctionThe Fourth-Root FunctionPower Functions xa, 0 < a < 1Slide 13Multiples of FunctionsSlide 15Logarithmic FunctionsThe Logarithmic Function f(x) = log2 xGrowth of the Logarithmic Functionf(x) = log2 x vs. g(x) = x1/nLogarithmic Functions vs. Power Functionsf(x) = x vs. g(x) = x log2 xf(x) vs. f(x) log2 xSlide 23Multiplication of FunctionsExponential FunctionsThe Exponential Function f(x) = 2xGrowth of the Exponential FunctionSlide 28Slide 29f(x) = 2x vs. Power Functions (Small Values of x)f(x) = 2x vs. Power Functions (Large Values of x)Slide 32Real-Valued Functions of a Real Variable and Their GraphsLecture 38Section 9.1Mon, Mar 28, 2005FunctionsWe will consider real-valued functions that are of interest in studying the efficiency of algorithms.Power functionsLogarithmic functionsExponential functionsPower FunctionsA power function is a function of the formf(x) = xafor some real number a.We are interested in power functions where a  0.The Constant Function f(x) = 12 4 6 8 100.511.52The Linear Function f(x) = x2 4 6 8 10246810The Quadratic Function f (x) = x22 4 6 8 1020406080100The Cubic Function f(x) = x32 4 6 8 10100200300400500600Power Functions xa, a  1The higher the power of x, the faster the function grows.xa grows faster than xb if a > b.The Square-Root Function2 4 6 8 100.511.522.53The Cube-Root Function2 4 6 8 100.511.52The Fourth-Root Function2 4 6 8 100.250.50.7511.251.51.75Power Functions xa, 0 < a < 1The lower the power of x (i.e., the higher the root), the more slowly the function grows.xa grows more slowly than xb if a < b.This is the same rule as before, stated in the inverse.0.5 1 1.5 21234Power Functionsx3x2xxMultiples of Functions1 2 3 42.557.51012.515x2x2x3xMultiples of FunctionsNotice that x2 eventually exceeds any constant multiple of x.Generally, if f(x) grows faster than cg(x), for any real number c, then f(x) grows “significantly” faster than g(x).In other words, we think of g(x) and cg(x) as growing at “about the same rate.”Logarithmic FunctionsA logarithmic function is a function of the formf(x) = logb xwhere b > 1.The function logb x may be computed as (log10 x)/(log10 b).The Logarithmic Function f(x) = log2 x10 20 30 40 50 60-2246Growth of the Logarithmic FunctionThe logarithmic functions grow more and more slowly as x gets larger and larger.f(x) = log2 x vs. g(x) = x1/n5 10 15 20 25 30-224log2 xx1/2x1/3Logarithmic Functions vs. Power FunctionsThe logarithmic functions grow more slowly than any power function xa, 0 < a < 1.f(x) = x vs. g(x) = x log2 x0.5 1 1.5 2 2.5 31234xx log2 xf(x) vs. f(x) log2 xThe growth rate of log x is between the growth rates of 1 and x.Therefore, the growth rate of f(x) log x is between the growth rates of f(x) and x f(x).2 4 6 81020304050f(x) vs. f(x) log2 xx2x2 log2 xx log2 xxMultiplication of FunctionsIf f(x) grows faster than g(x), then f(x)h(x) grows faster than g(x)h(x), for all positive-valued functions h(x).If f(x) grows faster than g(x), and g(x) grows faster than h(x), then f(x) grows faster than h(x).Exponential FunctionsAn exponential function is a function of the formf(x) = ax,where a > 0.We are interested in power functions where a  1.The Exponential Function f(x) = 2x1 2 3 42.557.51012.515Growth of the Exponential FunctionThe exponential functions grow faster and faster as x gets larger and larger.The Exponential Function f(x) = 2x1 2 3 4204060802x3x4xGrowth of the Exponential FunctionThe higher the base, the faster the function growsax grows faster then bx, if a > b.f(x) = 2x vs. Power Functions (Small Values of x)0.5 1 1.5 2123452xf(x) = 2x vs. Power Functions (Large Values of x)5 10 15 205001000150020002500300035002xx3Growth of the Exponential FunctionEvery exponential function grows faster than every power function.ax grows faster than xb, for all a > 1, b >


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