Use these Inter 1st Year Maths 1A Formulas PDF Chapter 1 Functions to solve questions creatively.

## Intermediate 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 a e A, there exists a unique element b e 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. The set A is called the domain of f and B is called co-domain off.

Illustration: Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25}. Consider the relation f(x) = x^{2}, then f(1) = 1, f(2) = 4, f(3) = 9, f(4) = 16. Clearly each element in A has unique image in B. So, f: A → B.

f = {(1, 1) (2, 4) (3, 9) (4, 16)} is a function from A to. B.

Clearly Domain (f) = {1,2, 3, 4} and Range (f) = {1, 4, 9, 16}

Note : The number of functions that can be defined from, a non-empty finite set A into a non-empty finite set B is [n(B)]^{n(A)}.

If f: A → B a function then f(A) = {f(a) / a ∈ A} is called the range off. It is a subset of B (ie) f(A) ≤ B.

**One – one function (Injection):**

Let f: A B, then f is said to be one-one function if different elements of A have different f- images in B. Thus f: A → B is one-one. (μ) f: A B is an injection <=> ay a2 e A and f(a7) = f(aj implies that a_{1} = a_{2}

Illustration:

LetA = {1, 2, 3} and B = {2, 4, 6}. Consider f: A → B. f(x) = 2x then f(1) = 2, f(2) = 4 and f(3) = 6. Clearly f is a function from A to B such that different elements in A have different f – images in B.

∴ f: {(1, 2) (2, 4) (3, 6)} is one – one.

Note : The number of one-one functions that can be defined from a non-empty finite set A into a non-empty finite set B is n(B) P_{n(A)} if n(B) ≥ n(A) and zero if n(B) < n(A).

**Onto function (Surjection):**

Let f: A → B. If every element of B occurs as the image of at least one element of in A, then fis an onto function. Thus f: A → B is surjection iff for each b e B, 3 a e A such that f(a) = b. Clearly fis onto ⇔ Range (f) = B.

Illustration; Let A = {-1, 7, 2, -2}, B = {1, 4} and Let f: A → B, be a function defined by f(x) = x^{2} then fis onto because f(A) = {f(-1), f(1), f(2), f(-2)} = {1, 4} = B.

Note : The number of onto functions that can be defined from a non – empty finite set A onto a two element set B is 2^{n(A)} – 2 if n(A) ≥ 2 and zero if n(A) < 2.

**Bijective function :**

A function f: A → B is a bijection if

- It is one – one i.e., f(a) – f(b) ⇒ a = b ∀ a, b ∈ A.
- It is onto i.e., ∀ b ∈ B ∃ a ∈ A such that f(a) = b.

Note : Number of bijections that can be defined from A to B is [n(A)]!, [n(A) = n(B)].

If f: A → B is a bijection then the relation f^{-1} = {(b, a) / (a, b) ∈ f} is a function from B to A and is called the inverse function off.

**Constant function :**

Let f: A B defined in such a way that all the elements of A have the same f- image in B, then f is said to be a constant function.

Illustration : Let A – {1, 2, 3} and B = {6, 7, 8}. Let f: A → B

f(x) = 6 ∀ x ∈ A i.e., f = {(1, 6) (2, 6) (3, 6)} is a constant function.

The range of a constant function is a singleton set.

**Identity function :**

Let A be a non-empty set then the function f: A → A defined by, f(x) = x ∀ x ∈ I_{A} is called the identity function on A and is denoted by I_{A}. The identity function is bijective.

Let f: A → B, g: B → C be functions. Then gof: A → C is a function and (gof) (a) = g If (a)] ∀ a ∈ A, is called composite of ‘g’ with ‘f.

If f : A → B, g B → C are bijections so is (go f) : A → C and (gof)^{-1} = f^{-1}o g^{-1}.

- If f: A → B is a bijection, then fof
^{-1}= I_{B}and f^{-1}of = I_{A}. - If f: A → B, g: B → C such that go f = I
_{A}fog = I_{B}then f is a bijection and g = f^{-1}

Let A be a non-empty subset of R such that – x ∈ A, for all x ∈ A and f: A → R.

- If f(-x) = f(x), ∀ x ∈ A then fis called an EVEN function.
- If f(-x) = – f(x), ∀ x ∈ A then f is called an ODD function.

**Functions:**

Def 1:

A relation f from a set A into a set B is said to be a function or mapping from A into B if for each x ∈ A there exists a unique y ∈ B such that (x, y) ∈ f. It is denoted

by f : A → B.

Note: Example of a function may be represented diagrammatically. The above example can be written diagrammatically as follows.

Def 2:

A relation f from a set A into a set B is a said to be a function or mapping from a into B if

i) x ∈ A ⇒ f (x) ∈ B

ii) x_{1}, x_{2} ∈ A, x_{2} ⇒ f (x_{1}) = f (x_{2})

Def 3:

If f : A → B is a function, then A is called domain, B is called codomain and f (A) = {f (x): x ∈ A} is called range of f.

Def 4:

A function f : A → B if said to be one one function or injection from A into B if different element in A have different f-images in B.

Note:

- A function f : A → B is one one if f(x
_{1}, y) ∈ f,(x_{2}, y) ∈ f ⇒ x_{1}= x_{2}. - A function f : A → B is one one iff x
_{1}, x_{2}∈ A, x_{1}≠ x_{2}⇒ f (x_{1}) ^ f (x_{1}) - A function f : A → B is one one iff x
_{1}, x_{2}∈ A, f (x_{1}) = f (x_{2}) ⇒ x_{1}= x_{2} - A function f : A → B which is not one one is called many one function
- If f : A → B is one one and A, B are finite then n(A) < n(B).

Def 5:

A function f : A → B is said to be onto function or surjection from A onto B if f(A) = B.

Note:

- A function f : A → B is onto if y e B U ⇓ ∃x ∈ A ∋ f (x) = y .
- A function f : A → B which is not onto is called an into function.
- If A, B are two finite sets and f : A → B is onto then n(B) ≤ n(A).
- If A, B are two finite sets and n (B) = 2, then the number of onto functions that can be defined from A onto B is 2
^{n( A)}– 2.

Def 6:

A function f : A → B is said to be one one onto function or bijection from A onto B if f : A → B is both one one function and onto function.

Theorem: If f : A → B, g : B → C are two functions then the composite relation gof is a function a into C.

Theorem: If f : A → B, g : B → C are two one one onto functions then gof : A → C is also one one be onto.

i) Let x_{1}, x_{2} ∈ A and f (x_{1}) = f (x_{2}).

x_{1},x_{2} ∈ A, f : A → B ⇒ f (x_{1}), f (x_{2}) ∈ B

f (x_{1}), f (x_{2}) ∈ B → C, f (x_{2}) ⇒ g[f (x_{1})] = g[f (x_{2})] ⇒ (gof)(x_{1}) = (gof)(x_{2})

x_{1}, x_{2} ∈ A,(gof)(x_{1}) = (gof): A → C is one one ⇒ x_{1} = x_{2}

x_{1}, x_{2} ∈ A, f (x_{1}) = f (x_{2}) ⇒ x_{1} = x_{2}.

∴ f: A → B Is one one.

ii) Proof: let z ∈ C,g : B → C is onto B y ∈ B ∃:g (y) = z y ∈ Bf : A → B is onto

∃x ∈ A ∋ f (x) = y

G {f(x)} = t

(g o f) x = t

∀ z ∈ CB x ∈ A ∋ (gof)(x) = z.

∴ g is onto.

Def 7:

Two functions f : A → B, g : C → D are said to be equal if

- A = C, B = D
- f (x) = g (x) ∀ x ∈ A. It is denoted by f = g

Theorem:

If f : A → B, g : B → C , h: C → D are three functions, then ho(gof) = (hof )of

Theorem:

if A is set, then the identify relation I on A is one one onto.

Def 8:

If A is a set, then the function I on A defined by I(x) = x ∀ x ∈ A, is called identify function on A. it is denoted by IA.

Theorem: If f : A → B and IA, IB are identify functions on A, B respectively then

foIA = IBof = f .

Proof:

I_{A}: A → A , f: A → B ⇒ foI_{A}: A → B

f : A → B , I_{B} : B → B ⇒ I_{B}of: A → B

(foIA)(x) = f {IA(x)} = f (x), ∀x ∈ A ∴ f0I_{A} = f

(IBof)(x) = IB{f (x)} = f (x), ∀ x ∈ A ∴ I_{B}of = f

∴ foI_{A} = I_{B}of = f

Def 9:

If f : A → B is a function then {(y, x) ∈ B × A:(x, y) ∈ f} is called inverse of f. It is denoted by f^{-1}.

Def 10:

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

Theorem:

If f : A → B is a bijection, then f^{-1} of = IA, fof^{-1} = IB

Proof:

Since f : A → B is a bijection f^{-1}: B → A is also a bijection and

f^{-1} (y) = x ⇔ f (x) = y ∀ y ∈ B

f : A → B, f^{-1}: B → A ⇒ f^{-1} of: A → A

Clearly I_{A} : A → A such that I_{A} (x) = x, ∀ x ∈ A.

Let x ∈ A

x ∈ A, f : A → B ⇒ f (x) ∈ B

Let y = f(x)

y = f (x) ⇒ f^{-1}(y) = x

(f -1 of)(x) = f ^{-1}[ f (x) = f^{-1}( y) = x = I_{A} (x)

(f^{-1} of) (x) = IA (x) ∀ x ∈ A f^{-1}of = I_{A}

f^{-1}: B → A, f: A → B ⇒ fof^{-1}: B → B

Clearly I_{B} : B → B such that I_{B} (y) = y ∀ y ∈ B

Let y ∈ B

y ∈ B, f^{-1}: B → A = f^{-1}(y) ∈ A

Let f^{-1}(y) = x

f^{-1}(y) = x ⇒ f (x) = y

(fof’)(y) = f [ f ^{-1}( y)] = f (x) = y = I_{B} (y)

∴ (fof^{-1})(y) = I_{B} (y) ∀ y ∈ B

∴ fof^{-1} = I_{B}

Theorem: If f : A → B, g : B → C are two bijections then (gof )^{-1} = f^{-1}og^{-1}.

Proof:

f : A → B, g : B →C are bijections gof: A → C is bijection (gof )^{-1}: C → A is a bijection.

f : A → B is a bijection f^{-1}: B → A is a bijection

g : B → C Is a bijection ⇒ g^{-1}: C → B is a bijection

g^{-1}:C → B , g^{-1}: B → A are bijections ⇒ f^{-1} og^{-1}: C → A is a bijection

Let z ∈ C

z ∈ C, g : B → C is onto ⇒ ∃ y ∈ B ∋ g (y) = z ⇒ g^{-1}(z) = y

y e B, f: A → B is onto ⇒ ∃ x ∈ A ∋ f (x) = y ⇒ f^{-1}(y) = x

(gof) (x) = g[ f (x)] = g (y) = z ⇒ (gof )^{-1}(z) = x

∴ (gof)^{-1} (z) = x = f^{-1}( y) = f^{-1} [ g^{-1} (z) ] = (f ^{-1}og^{-1})(z)

∴ (gof )^{-1} = f^{-1}og^{-1}

Theorem:

If f : A → B, g : B → A are two functions such that gof = I_{A} and fog = I_{B} then f : A → B is a bijection and f^{-1} = g .

Proof:

Let x_{1}, x_{2} ∈ A, f (x_{1}) = f (x_{2})

x_{1}, x_{2} ∈ A, f : A → B ⇒ f (x_{1}), f (x_{2}) ∈ B

f (x_{1}), f (x_{2}) ∈ B, f (x_{1}) = f (x_{2}), g = B → A

⇒ g [ f (x_{1})] = g[ f (x_{2})]

⇒ (gof)(x_{1}) = (gof)(x_{1}) ⇒ I_{A} (x_{2}) ⇒ x_{1} = x_{2}

x_{1},x_{2} ∈ A, f (x_{1}) = f (x_{2}) ⇒ x_{1} = x_{2}.

∴ f : A → B is one one

Let y ∈ B .

y ∈ B, g : B → A ⇒ g(y) ∈ A

Def 11:

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

Def 12:

A function f : A → B is said to be a real variable function if A ⊆ R.

Def 13:

A function f : A → B is said to be a real valued function iff B ⊆ R.

Def 14:

A function f : A → B is said to be a real function if A ⊆ R, B ⊆ R.

Def 15:

If f : A → R, g : B → R then f + g : A ∩ B → R is defined as (f + g)(x) = f (x) + g (x) ∀ x ∈ A ∩ B

Def 16:

If f : A → R and k e R then kf : A → R is defined as (kf)(x) = kf (x), ∀ x ∈ A

Def 17:

If f : A → B, g : B → R then fg : A n B → R is defined as (fg)(x) = f (x)g(x) ∀ x ∈ A ∩ B .

Def 18:

If f : A → R, g : B → R then : C → R is defined as

C = {x ∈ A n B: g(x) ≠ 0}.

Def 19:

If f : A → R then |f| (x) =| f (x)|, ∀ x ∈ A

Def 20:

If n ∈ Z , n ≥ 0, a_{0}, a_{1}, a_{2}, ………….. a_{n} ∈ R, a_{n} ≠ 0, then the function f : R → R defined by

f (x) = a_{0} + a_{1}x + a_{2}x^{2} + + a_{n}x^{n} ∀ x ∈ R is called a polynomial function of degree n.

Def 21:

If f : R → R, g : R → R are two polynomial functions, then the quotient f/g is called a rational function.

Def 22:

A function f : A → R is said to be bounded on A if there exists real numbers k1, k2 such that k1 < f (x) < k2 ∀ x ∈ A Def 23: A function f : A → R is said to be an even function if f (-x) = f (x) ∀ x ∈ A Def 24: A function f : A → R is said to be an odd function if f (-x) = – f (x) ∀ x ∈ A . Def 25: If a ∈ R, a > 0 then the function f : R → R defined as f (x) = ax is called an exponential function.

Def 26:

If a ∈ R, a > 0, a ≠ 1 then the function f : (0, ∞) → R defined as f (x) = log_{a} x is called a logarithmic function.

Def 27:

The function f : R → R defined as f(x) = n where n ∈ Z such that n ≤ x < n + 1 ∀ x ∈ R is called step function or greatest integer function. It is denoted by f (x) = [x]

Def 28:

The functions f(x) = sin x, cos x, tan x, cot x, sec x or cosec x are called trigonometric functions.

Def 29:

The functions f (x) = sin^{-1} x ,cos^{-1}x,tan^{-1} x,cot^{-1}x,sec^{-1}x or cos ec^{-1} x are called inverse trigonometric functions.

Def 30:

The functions f(x) = sinh x, cosh x, coth x, sech x or cosech x are called hyperbolic functions.

Def 30:

The functions f(x) = sinh^{-1}x, cosh^{-1}x, coth^{-1}x, sech^{-1}x or cosech^{-1}x are called Inverse hyperbolic functions.

Function | Domain | Range |

ax | R | (0, ∞) |

log a x | (0, ∞) | R |

[X] | R | Z |

[X] | R | [0, ∞) |

√ x | [0, ∞) | [0, ∞) |

sin x | R | [-1, 1] |

cos x | R | [-1, 1] |

tan x | R – {(2n +1)\(\frac{\pi}{2}\): n ∈ Z} | R |

cot x | R – {nπ: n ∈ Z} | R |

sec x | R – {(2n +1)\(\frac{\pi}{2}\): n ∈ Z} | (-∞,-1] ∪ [1, ∞) |

cos ecx | R – {nπ: n ∈ Z} | (-∞,-1] ∪ [1, ∞) |

Sin^{-1}x |
[-1 , 1] | [-π/2, π/2] |

Cos^{-1}x |
[ -1, 1] | [0, π] |

Tan^{-1}x |
R | (-π/2, π/2) |

Cot^{-1}x |
R | (0, π) |

Sec^{-1}x |
(-∞ -1] ∪ [1, ∞) | [0, π/2) ∪ (π/2, π] |

Cosec^{-1}x |
(-∞ -1] ∪ [1, ∞) | [-π/2,0) ∪ (0, π/2] |

sinh x | R | R |

cosh x | R | [1, ∞) |

tanh x | R | (-1,1) |

coth x | (-∞,0) ∪ (0, ∞) | (-∞,-1) ∪ (1, ∞) |

sech x | R | (0, 1] |

cosech x | (-∞,0) ∪ (0, ∞) | (-∞,0) ∪ (0, ∞) |

Sinh^{-1}x |
R | R |

Cosh^{-1}x |
[1, ∞) | [0, ∞) |

Tanh^{-1}x |
(-1, 1) | R |

Coth^{-1}x |
(-∞,-1) ∪ (1, ∞) | (-∞,0) ∪ (0, ∞) |

Sech^{-1}x |
(0, 1] | [0, ∞) |

Coseh^{-1}x |
(-∞,0) ∪ (0, ∞) | (-∞,0) ∪ (0, ∞) |