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, 2005FunctionsWe will consider real-valued functions that are of interest in studying the efficiency of algorithms.Power functionsLogarithmic functionsExponential functionsPower FunctionsA 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 1The 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 < 1The 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 Functionsx3x2xxMultiples of Functions1 2 3 42.557.51012.515x2x2x3xMultiples of FunctionsNotice 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 FunctionsA 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 FunctionThe 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 FunctionsThe 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 xThe 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 FunctionsIf 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 FunctionsAn 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 FunctionThe exponential functions grow faster and faster as x gets larger and larger.The Exponential Function f(x) = 2x1 2 3 4204060802x3x4xGrowth of the Exponential FunctionThe higher the base, the faster the function growsax 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 FunctionEvery exponential function grows faster than every power function.ax grows faster than xb, for all a > 1, b >
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