Lines and Functions

Lines and Slope

Recall that the slope of a line through points (x1,y1) and (x2,y2) is the rise over the run or

                  y2 - y1
      m  =                    
                   x2 - x1

For a function f(x) the secant line between x = a and x = b is the line through the points (a,f(a)) and (b,f(b))

 

Example  Finding the slope of the secant line

Find the equation of the secant line for y = x2 between x = 1 and x = 3.

 

Solution

At x  =  1,

        f(1)  =  12  =  1

and at x = 2, 

        f(2)  =  22  =  4

We need the equation of the line through the points (1,1) and (2,4).  The slope of the line is 

                  4 - 1
      m  =                  =  3
                  2 - 1

If we know that the slope of a line is m and the line passes through the point (x1,y1) then the equation of the line can be found by the using the formula:

        y - y1 = m(x - x1)

We have 

        y - 1 = 3(x - 1)  =  3x - 3

        y  =  3x - 2        adding 1 to both sides

 

Two lines can be parallel , perpendicular, or neither.  

If they are parallel, then they have the same slope:  m1 = m2  

they are perpendicular if the slopes are negative reciprocals of each other:  m1 = -1/m2  

Example

The lines 

        y  =  3x - 4        and        y  =  3x + 7 

are parallel since they have the same slope.

and the lines 

        y  =  -5x + 2    and    y  =  1/5 x - 1

are perpendicular since if we multiply the slopes together, the product is -1.

        (-5)(1/5)  =  -1

 

 

Functions

Recall that a function is a rule that assigns to each element of one set called the domain a unique element of another set called the range.

 

Example

Blood Pressure, the domain is the set of people and the range is the set of possible blood pressures.

 

Composition of Functions

Example

If 

        f(x)  =  3x + 1     and     g(x)  =  2x - 1 

Find 

A.      f(g(x))

B.         f(x + 2) - f(x)
                                        
                      2

 

Solution

A.        f(g(x))  =  3(2x - 1) + 1  =  6x - 2

 

B.  First notice that 

        f(x + 2)  =  3(x + 2) + 1

so that 

        f(x + 2)  - f(x)  =  3(x + 2) + 1 - (3x + 1)  =  3x + 6 + 1 - 3x - 1

        =  6
                

Exercise 

With g as in the example above find

            g(x + 3) - g(x)
                                        
                      3

 

Exercise:  Suppose that 

        f(x) = Pekt,   f(0) = 10,   f(1) = 25

find f(5)

Inverses:  If f passes the horizontal line test then f has an inverse.  

To find an inverse we

1)  switch x and y

2)  solve for y

 

Example

Find the inverse of 

                      x + 1
        f(x)  =              
                     2x - 5

 

Solution

We begin by switching the x's and y's.

                  y + 1
        x  =              
                 2y - 5

        x(2y - 5)  =  y + 1        Multiply by 2y - 5

        2xy - 5x  =  y + 1        Distribute the x through

        2xy - y  =  5x + 1        Bring the y terms to the left and the others to the right

        y(2x - 1)  =  5x + 1      Factor out a y

                5x + 1
        y  =                              Divide by 2x - 1
                 2x - 1

We can conclude that 

                       5x + 1
        f -1(x)  =                        
                        2x - 1

 

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