Can affine functions form a group under function composition?

Click For Summary
SUMMARY

The set A of affine functions defined as f(x) = mx + b, where m ≠ 0, forms a group under function composition. To prove this, one must demonstrate that the composition of any two affine functions results in another affine function within the set A. This involves verifying that the group axioms—closure, associativity, identity, and invertibility—are satisfied. The discussion concludes that by applying the composition property, the proof can be successfully completed.

PREREQUISITES
  • Understanding of function composition
  • Knowledge of group theory axioms
  • Familiarity with affine functions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of function composition in detail
  • Review group theory, focusing on the four group axioms
  • Explore examples of affine functions and their compositions
  • Learn about other types of functions and their group properties
USEFUL FOR

Mathematics students, particularly those studying abstract algebra, and anyone interested in the properties of functions and group theory.

POtment
Messages
28
Reaction score
0

Homework Statement


Show that the set A = (f:R-->R such that f(x)=mx+b, m not= 0} of affine functions from R to R forms a group under composition of function.

The Attempt at a Solution


Obviously I need to apply the composition of functions property (f: S->T, g:T->U, g of is function from S to U defined by g(f(x)) for all x element of S), but I'm not sure how to take the first step.
 
Physics news on Phys.org
Take two arbitrary elements of the group, compose them and show what you get must also be in the group.

That's one of the first steps. You need to show it satisfies all of the group axoims for the full proof.
 
Last edited:
Thanks, solved!
 

Similar threads

  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 5 ·
Replies
5
Views
1K
  • · Replies 9 ·
Replies
9
Views
2K
Replies
4
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
Replies
20
Views
5K
  • · Replies 15 ·
Replies
15
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K