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 - 1are 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.   ExampleBlood 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) + 1so 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 - 1We can conclude that                        5x + 1         f -1(x)  =                                                 2x - 1