Exponential

1
Chapter 4

• Exponential Logarithmic Functions

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4.1 Exponential Functions

• Objectives
• Evaluate exponential functions.
• Graph exponential functions.
• Evaluate functions with base e.

3
Definition of exponential function

• where b is a positive real number, which is
  not equal to 1. As, f(x) 1x is not exponential.
• Note that this is not the same as a power
  function. Observe the difference between
• f(x) 2x and g(x) x2.

4
Graph of an exponential function

• Graph
• As x values increase, f(x) grows RAPIDLY
• As x values become negative, with the magnitude
  getting larger, f(x) gets closer closer to
  zero, but with NEVER 0.
• f(x) is NEVER negative

5
Other characteristics of ______

• The domain of any elementary exponenctial
  functions is all real numbers.
• The range of any elementary exponential functions
  is the positive real numbers.
• The y-intercept is the point (0,1) (a non-zero
  base raised to a zero exponent 1)
• Every elementary exponential function has a
  horizontal asymptote y 0.
• If the base b lies between 0 1, the graph
  extends UP as you go left of zero, and gets VERY
  close to zero as you go right.
• Transformations of the exponential function are
  treated as transformation of polynomials (follow
  order of transformations as before)

6
Graph ____________

• First we must draw y2x
• Then shift the graph horizontally to the left by
  3 units, to get y 2x3.
• Finally, shift the graph vertically down by 4
  units, to get f(x).
• Note Point (0,1) has now been moved to
  (-3,-3) and the horizontal asymptote is y -4.

7
The special base e

• What is e? e is the so called Euler constant,
  which is an irrational number not unlike p. The
  value of e 2.718, so it is a number between 2
  and 3. This is what we call the natural base, as
  we are able to model many events in real life
  using exponential functions with base e.
• Lets graph f(x) ex and then f(x) -e-x

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Applications of exponential functions

• Exponential growth (compound interest!)
• Exponential decay (decomposition of radioactive
  substances)