~Composition of Functions~

~This is a way of combining 2 or more functions in a special way. Think of a function as a mapping that takes a point in one set (pre-image) to a point in another set (image). (i.e., set A to image points in set B) (x to f(x))

~The domain (accepted values) of the function f are the points in set A. A point in the domain of f, x, is taken from  set A to its image under f to f(x) in set B.

~The set of all images under f are points in set B. The set of these image points is the range of f.

~There might be points in set B that are not image points under f, thus not in the range of f.

~Let g be a function that takes image points in B to points in set C.
f(x) to g[f(x)], where f(x) is a point in set B & g[f(x)] is it's image point in set C.

~The function g takes points in the range of f (image points in set B) to their images in set C.

~It is necessary that the image of points under f (these are the f(x)’s) are accepted by g (in the domain of g). If not, they will not have images in set C & the composition is not defined.

~Ex: Let f(x) = -x²-1  and  g(x) = sqr(x). We see here that the
composition is not defined since all points in the range of f are not accepted by g. That is, the range of f consists of negative numbers and the domain of g are just positive numbers. We will only consider compositions that are defined.

~Ex: Let f(x) = x+3 and g(x) = x² . Find the function composition  (fog)(x). By definition, (fog)(x) = f [g(x)] . This means, replace x in function f by g(x).
Therefore, (fog)(x)= f[g(x)] = f[x²]= x²+3.

~Ex: If we reverse the composition and calculate (gof)(x) = g[f(x)] (this means, replace x in function g by f(x)), we get, (gof)(x) = g[f(x)] = g(x+3) = (x+3)².  We notice that (fog)(x) does not equal (gof)(x).
Generally, function composition IS NOT a commutative operation.

~Note: The beginning of a function compositon does not always start with an operation on a single variable. For example, to do the composition (gof)(3x-1) using the above functions, we need to calculate g[f(3x-1)]
= g[(3x-1)+3]=g(3x+2)=(3x+2)².  

1-1 Functions

~A function is 1-1, if for every image there is EXACTLY one pre-image. That is, for every value of f(x) there is exactly one value of x that
corresponds to that value. (for each y-value there is exactly one x-value).

~Ex:   f(x) = x² is not 1-1. Since f(2)=4 and (-2)=4. So, we have a y-value of 4 with two x-values.

~Ex:   f(x) = x³ is 1-1. You cannot have 2 different numbers giving the same result when cubed.

~Graphically, 1-1 functions satisfy the HORIZONTAL LINE TEST.  That is, all horizontal lines that you can draw that intersect the graph, must do so in EXACTLY ONE POINT.

~Ex:  f(x)=x⁴ (not 1-1),  g(x)=x³ (yes),   y=sqr(x)(yes),   y=2x+3  (yes)

~Note:   1-1 functions are very special, they have inverses.
~How to find the inverse of a 1-1 function. Use the procedure below.

  1.   Replace function notation with y-notation. (i.e., replace f(x) by y).
  2.   Interchange x and y.
  3.   Solve the new equation for y  (this is the inverse function).
  4.   Special notation is used for the inverse.  f^(-1)(x) is the symbol for the inverse of f.


~Ex:   If f(x) = 3x - 2, find f^(-1)(x), the inverse of f.  Note: f is 1-1, so f^(-1)(x) will exist

   1.   Replace f(x) by y:   y = 3x-2
   2.   Interchange  x and y:   x=3y-2
   3.   Solve for y:    x+2=3y gives (x + 2)/3 = y   ,  so, f^(-1)(x) = (x + 2)/3
                                                                                              


~Ex:   If f(x) = x³ , find f^(-1)(x).  Note: f is 1-1, so f^-1 will exist.

          1.   Replace f(x) by y:   y = x³   .
          2.   Interchange  x and y:  x = y³   
          3.   Solve for y:   y = x^(1/3)  (cube root of x)

~Note:   Inverses are very useful for finding pre-images if their images are known. (i.e., finding x, given f(x) or finding x, given y.)

Visualizing the inverse of a function

~Note:   If we graph a 1-1 function and its inverse on the same set of axes, we will notice that they are reflections of each other about the line  y = x. This reflection can easily be performed by folding a piece of paper.

~Note:   The line  y = x  is a line with slope of one passing through the origin. To get the sketch of the inverse with a piece of paper, sketch the curve for which the inverse function is desired. Make sure it is a 1-1 function or the inverse will not exist.  Also, do not label the axes.  Then, you must make a crease along this line. Then fold the top part of the crease down and bottom part of crease back and up. Do not rotate the paper. You will then be looking at the back of the paper at the graph of the inverse function.  Hold it up to a light source for a better view.