## Relations and Functions

# 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^{−1}og^{−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 | E^{n} = 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
^{−1}of = I_{A}and fof^{−1}= I_{B}. - If f : A → B is a bijection if and only if there exists a function g : B → A such that gof = I
_{A}.

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

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- 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
- Composite of functions is not commutative i.e., fog ≠ gof.
- Composite of functions is associative i.e., (fog)oh = fo(goh).
- If f : A → B and g : B → C are two bijections, then gof : A → C is bijection and (gof)
^{−1}= (f^{−1}og^{−1}). - fog ≠ gof but if, fog = gof then either f
^{−1}= g or g^{−1}= f also, (fog)(x) = (gof)(x) = (x). - (f
^{−1})^{−1}= f - If f and g are two bijections such that (gof) exists then (gof)
^{−1}= f^{−1}og^{−1}.