]]>
LearnNext
##### Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back

clear

# Theorems on Composition of Functions

5,471 Views
Have a doubt? Clear it now.
live_help Have a doubt, Ask our Expert
format_list_bulleted Take this Lesson Test

#### Theorems on Composition of Functions - Lesson Summary

Theorem 1 : If f : A â†’ B and g : B â†’ C are one-one, then gof : A â†’ C is also one-one.

Proof:
A function f : A â†’ B is defined to be one-one, if the images of distinct elements of A under f are distinct, i.e. for every x1, x2 âˆˆ A, f(x1) = f (x2) implies x1 = x2.

Given that f: A â†’ B and g: B â†’ C are one-one.

For any x1, x2 âˆˆ A

f(x1)=f(x2) â‡’ x1=x2 â€¦(i)

g(x1)=g(x2) â‡’ x1=x2 â€¦(ii)

To show: If gof(x1) = gof(x2), then x1 = x2

Let gof(x1) = gof(x2)

â‡’ g[f(x1)] = g[f(x2)]

â‡’ f(x1) = f(x2) â€¦from (i)

â‡’ x1 = x2 â€¦from (ii)

Hence, the functions gof: A â†’ C are one-one.

Theorem2: If f : A â†’ B and g : B â†’ C are onto, then gof : A â†’ C is also onto.

Proof:

Let us consider an arbitrary element z âˆˆ C

'.' g is onto âˆƒ a pre-image y of z under the function g such that g (y) = z â€¦â€¦â€¦(i)

Also, f is onto, and hence, for y Î B, there exists an element x âˆˆ A such that f (x) = y â€¦â€¦(ii)

Therefore, gof (x) = g (f (x)) = g (y) from (ii)

= z from (i)

Thus, corresponding to any element z âˆˆ C, there exists an element x âˆˆ A such that gof (x) = z.

Hence, gof is onto.

Note: In general, if gof is one-one, then f is one-one. Similarly, if gof is onto, then g is onto.

The composition of functions can be considered for n number of functions.

Theorem 3: If f : X â†’ Y, g : Y â†’ Z and h : Z â†’ S are functions, then ho(gof ) = (hog) o f.

Proof: Let x âˆˆ A

LHS: ho(gof ) (x)

= h(gof(x))

= h(g(f(x))), âˆ€ x in X

RHS: (hog) of f(x)

= hog(f(x))

= h(g(f (x))), âˆ€ x in X.

LHS = RHS

Hence, ho(gof) = (hog)of.

The composition of functions satisfies the associative property.