Function Algebra and Important Functions   Function notation We write f(x) to mean the function whose input is x.   Examples: If          f(x)  =  2x - 3 then          f(4)  =  2(4) - 3  =  5 We can think of f and the function that takes the input multiplies it by 2 and subtracts 3.  Sometimes it is convenient to write f(x) without the x. Thus:         f( ) = 2( ) - 3 whatever is in the parentheses, we put inside.  For example:         f(x - 1)  =  2(x - 1) - 3         f(x + 4) - f(x)              [2(x + 4) - 3] - [2(x) - 3]                                  =                                                                     4                                          4                 2x + 8 - 3 - 2x + 3                 8         =                                          =               =   2                                           4                               4 Exercises Find   f(2x + 1)          f(x2) - f(x)          Composition of Functions If f(x) and g(x) are functions then we define          f o g (x)   =    f(g(x))   Example: If          f(x) = 4/x  and          g(x) = x2 - x Then                                   4          f o g(x) =                                                        x2 - x  The Constant Function Example Let          f(x)   =  x - 1  and         k(x)  =  4 Then k(x) is the function that gives the number 4 as its output no matter what the input.  For example:         k(10)   =   4         f(k(x))   =   f(4)   =   3         k(f(x))   =   4     Function Arithmetic We define the sum, difference, product and quotient of functions in the obvious way. Example: If                        x + 1         f(x)  =                                           x - 1 and          g(x)  =  x2 + 4 then                                x + 1         (f + g)(x)  =                 +   (x2 + 4)                                 x - 1                                 x + 1         (f - g)(x)  =                  -   (x2 + 4)                                x - 1                               x + 1         (f g)(x)  =                (x2 + 4)                              x - 1                                  x + 1                                                                                x - 1          (f / g)(x)  =                                                          x2 + 4 For an interactive demonstration of the arithmetic of functions go to  This Link    Four Important Functions.There are four important functions and their variations that we discus here.  A list is given below:  Linear Functions (lines) You can identify a linear function as         f(x) = mx + b their graphs can be found by plotting the y intercept and using the slope to plot the second point.  To learn more about lines go to Functions and Graphs Quadratic functions (parabolas) You can identify a quadratic function by the equation         f(x) = ax2 + bx + c Their pictures are parabolas.  Previously, we discussed how to graph a parabola in-depth.  For a full review go to Parabolas   Square Root Functions You can identify a square root function by the equation         f(x)   =   They are half parabolas on their sides.   Absolute Value functions: You can identify  an absolute value function by the equation         y   =   |x| These look like the letter "V".   Example:   The graph of the function          f(x)   =   |x - 2|  + 1 Notice that the vertex is at the point (2,1) since the graph is shifted right 2 and up 1.           A polynomial  function is a sum of multiples of powers of x.  For example,         f(x) = 3x4 - 2x3 +  x2 - 3 Their graphs are beyond the scope of this course.     Back to the Intermediate Algebra (Math 154) Home Page e-mail Questions and Suggestions