# Composition of Functions and Invertible Function

• Composite function: If  f and g are two functions, then the composite function f and g is denoted by gof and it is defined only when codomain of f equal to domain of g. i.e.
• If f : A → B and g : B → C are two functions, then the composite function gof : A → C is defined as gof(x) = g[f(x)] ∀ x ∈ A.
• If f and g are two bijective functions then gof is also bijective.
• If gof is one one, then f is one one but g may not be one one.
• If gof is onto, then g is onto but f may not be onto.
• Inverse of a function: If the function f : x → y is bijective, then we defined inverse of f as f−1 : y → x by the rule y = f(x) ⇔ x = f−1 (y) ∀ x ∈ X, y ∈ Y.
• If f : A → B and g : B → C are two bijective functions then (gof)−1 = f−1og−1
• The graph of y = f(x) and y = f−1(x) intersects on the line y = x i.e the solution of f(x) = f−1(x) is solution of y = x
• Even and odd function: A function f(x) is said to be f(−x) = f(x) then it even function.
A function f(x) is said to be f(−x) = −f(x), then it is odd function.
• Every function f(x) can be express as sum of even and odd functions. ie f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}
• The graph of even function is symmetric about the y-axis.
• The graph of odd function is symmetric about the origin and is placed in first and third quadrant (or) second and fourth quadrants.
• If f(x) is even function, then f'(x) is odd function provided f'(x) is defined on R.
• If f(x) is odd function, then f'(x) is even function provided f'(x) is defined on R.

1.

 f g fog gof Even Even Even Even Even Odd Even Even Odd Even Even Even Odd Odd Odd Odd

2. E = Even function, O = Odd function, N = Neither even nor odd function

 E + E = E E − E = E O − O = O E E = E O + O = O E − O = O − E = N O O = E \frac{E}{E}=E En = E E O = O E = O \frac{O}{E}=\frac{E}{O}=O\frac{O}{O}=E O^{n}=\begin{cases} O &\tt n \ is \ odd\\E & \tt n \ \ is \ even\end{cases}
• All even functions are many one.
• Even extension: The even extension is obtained by defining a new function g(x) as
g(x)=\begin{cases}f(x) & x \in [0, a]\\f(-x) & x \in [-a, 0]\end{cases}
• Odd extension: The odd entension is obtained by defining a new function g(x) as
g(x)=\begin{cases}f(x) & x \in [0, a]\\-f(-x) & x \in [-a, 0]\end{cases}
• If f : A → B is a bijection, then f−1 : B → A is also a bijective function.
• If f : A → B is a bijection, then (f−1)−1 = f
• The inversion of a bijection is unique
• If f : A → B is a bijection then f−1of = IA and fof−1 = IB.
• If f : A → B is a bijection if and only if there exists a function g : B → A such that gof = IA.

### View the Topic in this video From 20:35 To 55:26

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1. If f : A → B and g : B → C are two function then the composite function of f and g, gof A → C will be defined as gof(x) = g[f(x)], ∀ x ∈ A
2. Composite of functions is not commutative i.e., fog ≠ gof.
3. Composite of functions is associative i.e., (fog)oh = fo(goh).
4. If f : A → B and g : B → C are two bijections, then gof : A → C is bijection and (gof)−1 = (f−1og−1).
5. fog ≠ gof but if, fog = gof then either f−1 = g or g−1 = f also, (fog)(x) = (gof)(x) = (x).
6. (f−1)−1 = f
7. If f and g are two bijections such that (gof) exists then (gof)−1 = f−1og−1.