## Relations and Functions

# Types of Functions

**TYPES OF FUNCTIONS:**

- A function f: A → B is said to be one-one function (or) Injective function "if distinct elements is set A have distinct images in set B" i.e. no two elements in set A have same image in set B.
- The graph of the curve said to be one-one function "if the horizontal line cuts the graph at exactly one point".
- A function f: A → B is said to be one-one function if ∀ x
_{1}, x_{2}∈ A f(x_{1}) = f(x_{2}) ⇒ x_{1}= x_{2}(or) x_{1}≠ x_{2}⇒ f(x_{1}) ≠ f(x_{2}) - A function f: A → B is said to be one-one function "if it is either strictly increasing f’(x) > 0 (or) strictly decreasing function f”(x) < 0".
- A function f: A → B is not one-one function then it is said be many one function.
- The number of one-one functions that can be defined from the set A to the set B is = \begin{cases}{^{n(B)}P_{n(A)}}; & if \ n(A) \leq n(B)\\0 \ ; & if \ n(A) \geq n(B)\end{cases}
- A function f: A → B is said to be onto function (or) surjective function "if every element in set B has at least one preimage in set A"
- The graph of the curve is said to be onto function "if the horizontal line in the codomain of the curve cuts at least one point".
- A function f: A → B is said to be onto function if ∀ y ∈ B ∃ x ∈ A such that f(x) = y
- A function f: A → B is said to be onto function if range of f = codomain of f.
- A function f: A → B is said to be not onto function then it is said to be not onto function then it is said to be into function.
- A function f : A → B is said to be bijective function "if it is both one one function and onto function" if n(A) = n(B).

**Short cut Tips:**

- log
_{b}a > k \Rightarrow \begin{cases}a > b^{k} & {\tt if} \ b > 1\\ a < b^{k} & {\tt if} \ b < 1 \end{cases} - \sqrt[n]{x^{n}}=\begin{cases}|x| &\tt n \ is \ even\\ x &\tt n \ is \ odd \end{cases}
- |x| ≤ a ⇒ − a ≤ x ≤ a

|x| ≥ a ⇒ x ≤ −a (or) x ≥ a

a ≤ |x| ≤ b ⇒ x ∈ [−b, −a] ∪ [a, b] - The graph of the function y = f(x) is symmetric about the line x = k is f(k + x) = f (k − x)
- \left[\frac{1}{2}\right]+\left[\frac{1}{2}+\frac{1}{100}\right]+\left[\frac{1}{2}+\frac{2}{100}\right]+ ---- + \left[\frac{1}{2}+\frac{99}{100}\right]=50
- f : R → R is given by f(x) = \frac{a^{x}}{a^{x}+\sqrt{a}} ∀ x ∈ R then f\left(\frac{1}{1997}\right)+f\left(\frac{2}{1997}\right)+----+ f\left(\frac{1996}{1997}\right)=\frac{1996}{2}=998
- If f : R → R is given by f(x) + f(1 − x) = 1 then f\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)+----+ f\left(\frac{1995}{1996}\right)=\frac{1995}{2}

### View the Topic in this video From 08:20 To 20:30

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- The number of onto functions that can be defined from the set A to the set B if n(B) = 2 is 2
^{n(A)}− 2. - The number of bijective functions that can be defined from the set A to the set B is n(A)! (or) n(B)!
- The number of constant functions that can be defined from the set A to the set B is n(B).
- The number of many-one functions that can be defined from the set A to the set B is = n(B)
^{n(A)}–^{n(B)}P_{n(A)} _{The number of onto function that can be defined from the set A to the set B is =\begin{cases}\sum_{r=1}^{n} (-1)^{n-r} \ {^nC_r} \ r^{m}; & {\tt if} \ n(A) \geq n(B)\\0; & {\tt if} \ n(A) < n(B)\end{cases} where n(A) = m, n(B) = n.}