Learning these TS Inter 1st Year Maths 1A Formulas Chapter 1 Functions will help students to solve mathematical problems quickly.

## TS Inter 1st Year Maths 1A Functions Formulas

→ Function: Let A and B be non-empty sets and f be a relation from A to B. If for each element as A there exists a unique be B such that (a, b) ∈ f. then f is called a function (or) mapping from A to B (or A into B). It is denoted by f: A → B.

Eg : If A = {1, 2, 3}, B = {p, q, r}, f = {(1. p), (2, p), (3, p)} then f is a function from A to B.

If f: A → B is a function ∀ a ∈ A such that f(a) = b. 3 b ∈ B.

→ One-one function (or) Injection: A function f: A → B is said to be one-one function or injection from A into B if different elements in A have different T images in B. (March ’93)

Eg : If A = {1, 2, 3}, B = {p,q, r, s}, f – {(1, r), (2, p). (3, s)} then f: A → B is one – one. f: A → B is an injection

⇔ a_{1}, a_{2} ∈ A and a_{1} ≠ a_{2} ⇒ f(a_{1}) * f(a_{2})

⇔ a_{1}, a_{2} ∈ A and f(a_{1}) = f(a_{2}) ⇒ a_{1} = a_{2}

→ Onto function (or) surjection : A function f: A → B is said to be function (or) surjection from A onto B is f(A) = B. (or) If f: A B is a function, if every element of B occurs as the image of atleast one element of A then we say that f is an onto function (or) surjection or that f from A onto B.

Eg : If A = {1, 2, 3}, B = {p,q}, f = {(1. q), (2, p), (3, q)}, then f: A → B is onto, f: A → B is a surjection

⇔ range f – f(A) = B(codomain)

⇔ B = {f(a) / a ∈ A}

⇔ For every b e B there exists atleast one as A such that f(a) = b.

→ Bijection (or) one – one and onto function : A function f: A B is said to be one – one and onto function (or) bijection from A onto B. If f: A B is both one – one function and onto function.

Eg : If A = {1, 2, 3}, B = (p,q, r}, f = {(1, q), (2, r), (3, p)}, then f: A → B is one-one and onto, f: A → B is a bijection f is both one – one and onto

⇔ (i) If aj, a, e A and f(a_{1}) = f(a_{2}) ⇒ a_{1} – a_{2}

(ii) For every b e B there exists atleast one as A such that f(a) = b.

→ Equality of functions : Two functions f: A → B, g : A → B are said to be equal if f(x) = g(x). ∀ x ∈ A. It is denoted by f = g (or) let f and g be functions. We say f and g are equal and write f = g if domain of f equal to domain of g and f(x) = g(x),∀ x ∈ domain f.

→ Constant function: A function f: A → B is said to be a constant function if the range of T contains only one element i.e., f(x) = c ∀ x ∈ A where c is a fixed element of B.

Eg : A = {1, 2, 3, 4}, B = {a, b, c}, f = {(1, b), (2. b), (3, b), (4, b)}, then f is a constant function from A to B.

→ Identity function: If A is a non empty set then the function f: A A defined by f(x) = x, ∀ x ∈ A is eaiied the identity function on A and is denoted by I_{A}.

Eg : A = {1, 2, 3}, I_{A} = (1, 1), (2, 2), (3. 3)} The function on R defined as f(x) = x ∀ x ∈ R is the identity function on R.

→Inverse function : If f: A → B is a bijection then the function f’1 : B -4 A defined by f-1(y) = x. If f(x) = y, ∀ y ∈ B is called the inverse function of f.

Eg: Let A = {1, 2, 3}, B = {a, b, c} and f = {(1, a), (2. b), (3, c)} then the inverse function f^{-1} = {(a, 1), (b, 2), (c, 3)} and f^{-1}: B → A is also a bijection.

→ Composite function : If f: A → B, g : B → C are two functions then the function gof: A C defined by gof(x) = g[f(x)]. ∀ x ∈ A is called composite function f and g.

→ Even function : A function f: A → R is said to be an even function if f(-x) = f(x), ∀ x ∈ A.

Eg : f(x) = x^{2}, g(x) = cos x are even functions.

→ Odd function : A function f: A → R is said to be an odd function if f(-x) = – f(x), ∀ x ∈ A.

Eg : f(x)= x^{3}, g(x) = sin x are odd functions.

→ To Vind the domains of a Real valued functions :

- The domain of the real function is of the form \(\frac{1}{g(x)}\) (or) \(\frac{f(x)}{g(x)}\) is R – {x/g(x) = 0}
- The domain of the real function is of the form \(\sqrt{f(x)}\) is {x/f(x) ≥ 0}
- The domain of the real function is of the form \(\frac{1}{\sqrt{f(x)}}\) is {x/f(x) > 0}
- The domain of the real function is of the form log [f(x)] is {x/f(x) > 0}.

→ (i) (x – α)(x – β) < 0 ⇒ x ∈ [α, β].

(ii) (x – α) (x – β) < 0 ⇒ x ∈ (α, β).

(iii) (x – α) (x – β) > 0 ⇒ x ∈ R – (α, β) (or) x ∈ (- ∞. α] ∪ [β, ∞)

(iv) (x – α) (x – β) > 0 ⇒ x ∈ R – [α, β] (or) x ∈ (- ∞, α) ∪ (β, ∞)