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

1. f(2x + 1)         
   
   
   

2. 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:

1. 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
   
   

2. 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

    

3. Square Root Functions

   You can identify a square root function by the equation

           f(x)   =  

   They are half parabolas on their sides.

   

    

4. 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.  

          

Polynomials

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.  

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