Twelve Standard >> Composition of functions | composite function theorem and properties of composite function

Some interesting and important illustrations

Theorem 1: If f:A\(\rightarrow\) B and g:B\(\rightarrow\) C are one-one, then sho that gof:A\(\rightarrow\) C is also one-one.

Proof:
 Let \(x_1\), \(x_2\) \(\in\) A such that
   gof(\(x_1\))= gof(\(x_2\))
              [\(\because\) g is one-one
                 g\((y_1)\)=g\((y_20)\)
       (\Rightarrow\)  \((y_1)\)=\((y_2)\)    
                 f is one-one
                 f\((x_1)\)=g\((x_2)\)
                \(\Rightarrow\)  \((x_1)\)=\((x_2)\)]
\(\Rightarrow\) f(\(x_1\))=f(\(x_2\))
 \(\Rightarrow\) \(x_1\)=\(x_2\)
 \(\Rightarrow\) gof is one-one

Theorem 2:
Let f:A\(\rightarrow\) B and g:B(\Rightarrow\) C, are onto then prove that gof:A\(\rightarrow\) C is also onto.

Proof:

Let Z\(\in\) C be any arbitary element 
    as g is an onto function for all Z, then exists Y\(\in\) B such that f(y)=z
Since f is also onto, so for all Y\(\in\) B,  exist preimage x\(\in\) A such that f(x)=y
     gof(x)=g(y)=Z
\(\Rightarrow\) gof is an onto function

Some properties of composite function:

1) Composition function is associative, i.e f:A\(\rightarrow\) B, g:B\(\rightarrow\) C, h:C\(\rightarrow\) D are function then
  (i) ho(gof)=(hog)of

2) Also (f+g)oh=foh+goh

3) Let f:A\(\rightarrow\) B and g:B\(\rightarrow\) C be two functions then

   (i) If f and g are both one-one, then gof is also one-one.
   (ii) If f and g are both onto, then gof is also onto.
   But the converse is may not be true.