Real-Valued Functions of a Real Variable and Their Graphs

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Real-Valued Functions of a Real Variable and
Their Graphs

• Lecture 38
• Section 9.1
• Mon, Mar 28, 2005

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Functions

• We will consider real-valued functions that are
  of interest in studying the efficiency of
  algorithms.
• Power functions
• Logarithmic functions
• Exponential functions

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Power Functions

• A power function is a function of the form
• f(x) xa
• for some real number a.
• We are interested in power functions where a ? 0.

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The Constant Function f(x) 1
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The Linear Function f(x) x
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The Quadratic Function f(x) x2
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The Cubic Function f(x) x3
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Power Functions xa, a ? 1

• The higher the power of x, the faster the
  function grows.
• xa grows faster than xb if a gt b.

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The Square-Root Function
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The Cube-Root Function
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The Fourth-Root Function
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Power Functions xa, 0 lt a lt 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 lt b.
• This is the same rule as before, stated in the
  inverse.

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Power Functions
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Multiples of Functions
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Multiples 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.

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Logarithmic Functions

• A logarithmic function is a function of the form
• f(x) logb x
• where b gt 1.
• The function logb x may be computed as (log10
  x)/(log10 b).

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The Logarithmic Function f(x) log2 x
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Growth of the Logarithmic Function

• The logarithmic functions grow more and more
  slowly as x gets larger and larger.

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f(x) log2 x vs. g(x) x1/n
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Logarithmic Functions vs. Power Functions

• The logarithmic functions grow more slowly than
  any power function xa, 0 lt a lt 1.

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f(x) x vs. g(x) x log2 x
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f(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).

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f(x) vs. f(x) log2 x
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Multiplication 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).

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Exponential Functions

• An exponential function is a function of the form
• f(x) ax,
• where a gt 0.
• We are interested in power functions where a ? 1.

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The Exponential Function f(x) 2x
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Growth of the Exponential Function

• The exponential functions grow faster and faster
  as x gets larger and larger.

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The Exponential Function f(x) 2x
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Growth of the Exponential Function

• The higher the base, the faster the function
  grows
• ax grows faster then bx, if a gt b.

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f(x) 2x vs. Power Functions (Small Values of x)
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f(x) 2x vs. Power Functions (Large Values of x)
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Growth of the Exponential Function

• Every exponential function grows faster than
  every power function.
• ax grows faster than xb, for all a gt 1, b gt 0.