Chapter 5

Chapter 5

1. Exponential Growth and Decay

   • Growth: b>1
   • Decay: 0<b<1
   • Discrete V. Continuous
     • Discrete: Represented by separate points
     • Continuous: Represented by a curve or continuous line
   • Annual Growth and Decay
     • Negative Exponents
       • To go backwards in time
         • ex: 3^-2= 1/9
   • The Number e
     • Compounding growth and decay
     • This equation is only used when the phrase "continuously compounding" is used
       • • Logistic Growth: The rate of growth of a quantity may slow down after initially increasing or decreasing exponentially

2. Combining Functions

   • Function Operations
     • Addition
       • (f+g)(x)
     • Subtraction
       • (f-g)(x)
     • Multiplication
   • Composite Functions
     • Composite of functions f and g is f(g)(x)), f of g of x
       • Other notation
       • Find g(x) first, then plug that value into f(x) to solve
     • Domain and Range
       • Domain: All possible values of the independent variable (x)
       • Range: All possible values of the dependent variable (y)

3. Inverse Functions

   • If (a,b) is on the graph of a function, then (b,a) is on the graph of the inverse
   • Functions are inverse if f(a) = b and g(b) = a
   • The graph of the inverse is a reflections over the line y=x
   • Domain and Range are switched for its inverse
   • Notation
     • f^-1(x) or f^-1
   • Finding an inverse
     • The inverse must be a function in order to be the inverse; we are able to restrict the domain and range in order to make the inverse a function
       • Must also pass the vertical and horizontal line test
     • 1. Switch to "y" notation
       • 2. Swap x and y
         • 3. Solve for y

4. Fractional exponents

   • When solving remember radical properties

5. Logarithmic Functions

   • The inverse of an exponential function
   • Domain and Range
     • D: x>0
     • R: All real numbers
   • Common log and Natural Log
     • Common Log:
     • Natural Log:
   • Properties of Logarithms

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