what is necessary condition to find the composite function of two functions
Answers
Answer:
Function Composition" is applying one function to the results of another:
Function Composition
The result of f() is sent through g()
It is written: (g º f)(x)
Which means: g(f(x))
Example: f(x) = 2x+3 and g(x) = x2
"x" is just a placeholder. To avoid confusion let's just call it "input":
f(input) = 2(input)+3
g(input) = (input)2
Let's start:
(g º f)(x) = g(f(x))
First we apply f, then apply g to that result:
Function Composition
(g º f)(x) = (2x+3)2
What if we reverse the order of f and g?
(f º g)(x) = f(g(x))
First we apply g, then apply f to that result:
Function Composition
(f º g)(x) = 2x2+3
We get a different result!
When we reverse the order the result is rarely the same.
So be careful which function comes first.
Symbol
The symbol for composition is a small circle:
(g º f)(x)
It is not a filled in dot: (g · f)(x), as that means multiply.
Composed With Itself
We can even compose a function with itself!
Example: f(x) = 2x+3
(f º f)(x) = f(f(x))
First we apply f, then apply f to that result:
Function Composition
(f º f)(x) = 2(2x+3)+3 = 4x + 9
We should be able to do it without the pretty diagram:
(f º f)(x) = f(f(x))
= f(2x+3)
= 2(2x+3)+3
= 4x + 9
Domains
It has been easy so far, but now we must consider the Domains of the functions.
domain and range graph
The domain is the set of all the values that go into a function.
The function must work for all values we give it, so it is up to us to make sure we get the domain correct!
Example: the domain for √x (the square root of x)
We can't have the square root of a negative number (unless we use imaginary numbers, but we aren't), so we must exclude negative numbers:
The Domain of √x is all non-negative Real Numbers
On the Number Line it looks like:
zero onwards
Using set-builder notation it is written:
{ xmember ofreals | x ≥ 0}
Or using interval notation it is:
[0,+∞)
Answer:
Suppose f is a function which maps A to B and there is another function g (which may or may not be similar to f) which maps B to C. Now the question is can we map A to C? The mapping of elements of A to C is the basic concept of Composition of functions. When two functions combine in a way that the output of one function becomes the input of other, the function is a composite function.
In mathematics, the composition of a function is a step-wise application. For example, the function f: A→ B & g: B→ C can be composed to form a function which maps x in A to g(f(x)) in C. All sets are non-empty sets. A composite function is denoted by (g o f) (x) = g (f(x)). The notation g o f is read as “g of f”
Step-by-step explanation: The necessary condition to find the composition function of two function is that our inner outer functions domain should consist of the range of our inner function.