Composition Of Functions Implies Equality

In summary, the composition of functions is a mathematical operation that combines two functions to form a new function by using the output of one function as the input of the other. This operation implies equality between the composite function and the individual functions evaluated at the same input. The order of composition matters, as it can affect the resulting function, and not all functions can be composed. Composition of functions is used in various fields to model and analyze real-life situations.
  • #1
netcaster
4
0
We have three functions: f:A->A, g:A->A and h:A->A
with both f and g bijective and h bijective.

We know that f ° h = h ° g for every x in A.

Is it true that f=g for every x in A?

I have tried to solve it and I am pretty sure it is true but I can find neither a counterexample nor a simple proof.
 
Mathematics news on Phys.org
  • #2
Take A= {1, 2, 3}, f= {(1,3), (2,1), (3,2)}, g= {(1,2), (2,3), (3,1)}, h= {(1,3), (2,3), (3, 1)}.
 

Related to Composition Of Functions Implies Equality

1. What is the concept of composition of functions?

The composition of functions is a mathematical operation that combines two functions to form a new function. The output of one function becomes the input of the other function, resulting in a new function. This operation is denoted by (f ∘ g)(x) or f(g(x)).

2. How does composition of functions imply equality?

If two functions, f(x) and g(x), are composed to form a new function, (f ∘ g)(x) = f(g(x)). This implies that the output of the composite function is equal to the output of the individual functions evaluated at the same input.

3. What is the significance of the order of composition of functions?

The order of composition of functions matters because the resulting function may be different depending on the order in which the functions are composed. This is known as the commutative property of composition of functions.

4. Can any two functions be composed?

No, not all functions can be composed. In order for two functions to be composed, the output of the inner function must be a valid input for the outer function. This means that the domain of the inner function must be a subset of the domain of the outer function.

5. How is composition of functions used in real life?

Composition of functions is used in various fields, such as economics, engineering, and physics, to model and analyze real-life situations. For example, in economics, the total cost of producing a product can be represented as the composition of the cost of materials and the cost of labor.

Similar threads

MHB Compose Functions: True/False?
    • General Math
Replies
1
Views
991
I Best way to fit three functions
    • General Math
Replies
2
Views
752
A A bit of clarification on the domain of a composite function
    • General Math
Replies
5
Views
1K
B Proof of Chain Rule: Understanding the Limits
    • General Math
Replies
3
Views
840
I Can chatgpt accurately calculate expected lengths in Pascal's triangle?
    • General Math
2
Replies
66
Views
4K
I Why does the summation come from?
    • General Math
Replies
11
Views
1K
B Domain and the codomain of a composite function
    • General Math
Replies
8
Views
4K
I Alternating Harmonic Numbers are cool, spread the word!
    • General Math
Replies
4
Views
571
B Limits on Composite Functions- Appears DNE but has a limit
    • General Math
Replies
8
Views
3K
MHB Solving g(x)-h(x) <0: Find a, b, c, d
    • General Math
Replies
2
Views
903

Hot Threads

Recent Insights

Back
Top