Functions And Graphs

Functions And Graphing

1. Definition of a Function

    

   Definition A function is a rule that assigns every element from a set (called the domain) to a unique element of a set (called the range)

   
   
   Examples

   1. Let the domain be US citizens and the range be the set of all fathers.  Let the rule of the function send each person to his or her fathers.  

   2. f (x) = x²  

   3. f (x) = 3x + 1

   4. f (x) = 1/x  (the domain is R - {0})

   5. x² + y²  = 1  
      
      (not a function since for 
      
              x = 0
      
      y can be 1 or -1.

    

2. Vertical Line Test

   If every vertical line passes through the graph at most once, then the graph is the graph of a function.

   
    

3. Finding the Domain

   To find the domain of a function, we follow the three basic principals:

   1. The domain of a polynomial is R

   2. The domain of a rational function 
      
              p(x)/q(x) 
      
      is the set of all real numbers x such that 
      
             \( q(x) \ne 0 \)

   3. The domain of a square root function is all real numbers such that the function inside the square root is non-negative.  
      
      For example, to find the domain of 
      
             \( f(x) = \sqrt{4 - 2x} \)
      
      we just set
      
              4 - 2x > 0
      
      or 
      
              x < 2
      
      so that the domain of f(x) is 
      
              {x | x < 2}

   4.  For a word problem the domain is the set of all x values such that the problems makes sense.

   Examples:  

   The domain of 
   
          \( f(x) = \frac{x+1}{x-1} \)
   
   is all real numbers except where x = 1.
    

4. Function Evaluation

   
   If f is an algebraic function we can write without the variables as in the following example:

   If 
   
           f(x) = x² - 2x

   We can write

           f = (  )² - 2(  )  

   This more suggestively shows how to deal with non x inputs.  For example

           f(x - 1) = (x - 1)² - 2(x - 1)
    

5. The Difference Quotient

   We define the difference quotient for a function f by

          

   Difference Quotient              \[ \frac{f(x +h) - f(x)}{h}  \]

    

   Example
   
   Let 
   
           f(x) = x² - 2x
   
   Then

           f(x + h) = (x + h)² - 2(x + h) = x² + 2xh + h² -2x - 2h  
   
   and

           f(x) = (x)² - 2(x)

   so that
   
          \( \frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2) - (x^2 - 2x)}{h} \)
   
          \( = \frac{2xh + h^2 - 2h}{h} = \frac{h(2x + h - 2)}{h} = 2x + h - 2 \)
   
    

6. Finding the domain and range of a function from its graph.

   We often use the graphing calculator to find the domain and range of functions.  In general, the domain will be the set of all x values that has corresponding points on the graph.  We note that if there is an asymptote (shown as a vertical line on the TI 85) we do not include that x value in the domain.  To find the range, we seek the top and bottom of the graph.  The range will be all points from the top to the bottom (minus the breaks in the graph).  
    

7. Zeros of Functions and the x-intercept Method

   To find the x-intercepts of a complicated function, we can use the TI 85 to view the graph, then use 
   
           more math root 
   
   to find the x-intercepts.

   Exercise

   Find the roots of 

           y = x³ - 4x² + x + 2

   
   
   If we want to find the intercept of two graphs, we can set them equal to each other and then subtract to make the left hand side zero.  Then set the right hand side equal to y and find the zeros.

   Example:  

   Find the intercept of the graphs:

           y = 2x³ - 4 
   
   and

           y = x⁴ - x² 
    

   Solution:

   We form the new function
   
           f(x) = (2x³ - 4) - (x⁴ - x²)

    

   
   
   and use the root feature of our calculator to get
   
           x = 1.52    or    x = 2

    

8. Solving Inequalities Graphically 

   To solve an inequality graphically we first put 0 on the right hand side and f(x) on the left hand side.  Then we use the x-intercept method to find the zeros.  If the inequality is a "<" we include the part of the graph below the x axis.  If the inequality is a ">" we include the part of the graph above the x axis.
    

   Example:

   Give and approximate solution of
   
           3x⁵ - 14x²  <  x - 4
   
   
   Solution:

   First, bring everything to the left hand side
   
           3x⁵ - 14x² -  x + 4  <  0

    

   
   
   The graph shows that the roots lie at:
   
           x = -0.56,    x = 0.51,    and    x = 1.64
   
   The points lie below the x-axis to the left of -0.56 and between 0.51 and 1.64.  Hence the solution in interval notation is
   
          \( (-\infty,-0.56) \cup (0.51,1.64) \)
   
           

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