Function Formulas

The existing List of Function Formulas can make your calculations easy and helps you do your homework at a faster pace. Also, you can paste the function formulas cheat sheet & tables on your walls to memorize the formulas daily & master in solving the problems of Inverse function, Modulus function, Even and Odd Function, etc. Hence, refer to the below list of function formulae and get familiar with the concept of functions in a better way.

1. Numbers and their set

• Natural Numbers: N = {1, 2, 3, 4,…}
• Whole Numbers: W = {0, 1, 2, 3, 4,….}
• Integer Numbers: I or Z = {…-3, -2, -1, 0, 1, 2, 3,…}
  Z⁺ = {1, 2, 3,….}, Z^– = {-1, -2, -3,….}
  Z₀ = {± 1, ± 2, ± 3,….}
• Rational Numbers: Q = {p/q ; p, q eZ.q^ 0]
• Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q, called irrational numbers
  eg. {\(\sqrt{2}\), \(\sqrt{3}\), 2^1/3, 5^1/4 , π, e,….}

2. Interval

• Close interval [a, b] = {x, a ≤ x ≤ b}
• Open interval (a, b) or ]a, b[ = {x, a < x < b}
• Semi open or semi close interval
  [a, b[ or [a, b) = {x; a ≤ x < b}
  ]a, b] or (a, b] = {x; a < x ≤ b}

3. Function

Let A and B be two given sets and if each element a ∈ A is associated with a unique element b ∈ B under a rule f, then this relation is called Function.

Here, b is called the image of a and a is called the pre-image of b under f.
Domain = All possible values of x for which f(x) exists.
Range = For all values of x, all possible values of f(x).

Domain ={a, b, c, d} = A
Co-domain = {p, q, r, s} = B
Range = {p, q, r}

4. Even function

If we put (-x) in place of x in the given function and if f(-x) = f(x), ∀ x ∈ domain then function f(x) is called even function.

5. Odd function

If we put (-x) in place of x in the given function and if f(-x) = – f(x), ∀ x ∈ domain then f(x) is called odd function.

6. Properties of even and odd Function

• The product of two even functions is even function.
• The sum and difference of two even functions is even function.
• The sum and difference of two odd functions is odd function.
• The product of two odd functions is even function.
• The product of an even and an odd function is odd function.
• The sum of even and odd function is neither even nor odd function.

7. Explicit Function

A function is said to be explicit if it can be expressed directly in terms of the independent variable, y = f(x) or x = Φ(y)

8. Implicit Function

A function is said to be implicit if it can not be expressed directly in terms of the independent variable.
ax² + 2hxy + by² + 2gx + 2fy + c = 0

9. Increasing Function

A function f(x) is called increasing function in the domain D if the value of the function does not decrease by increasing the value of x. If x₁ > x₂ ⇒ f(x₁) > f(x₂) or x₁ < x₂ ⇒ f(x₁) < f(x₂) or f'(x) > 0 for increasing and f'(x) ≥ 0 for not decreasing.

10. Decreasing Function

A function f(x) is said to be decreasing function in the domain D if the value of the function does not increase by increasing the value of x (variable). If x₁ > x₂ ⇒ f(x₁) < f(x₂) or x₁ < x₂ ⇒ f(x₁) > f(x₂) or f'(x) < 0 for decreasing and f'(x) ≤ 0 for not increasing.

11. Greatest Integer Function A function is said to be greatest integer function if it is of the form of ffx) = [x] where [x] = integer equal or less than x. i.e. [4.2] = 4, [- 4.4] = – 5

Properties of G.I.F.:

• [x] = x if x is integer
• [x + I] = [x] + I, if I is an integer
• [x + y] ≥ [x] + [y]
• If [Φ(x)] ≥ I then Φ(x) > I
• If [Φ(x)] ≤ I then Φ(x) < I + 1
• If [x] > n ⇒ x ≥ n + 1
• If [x] < n ⇒ x < n, n ∈ I
• [-x] = – [x] if ∀ x ∈ I
• [-x] = – [x] – 1 if x ∉ Integer
• [x + y] = [x] + [y + x – [x]] ∀ x, y ∈ R
• [x] + \(\left[x+\frac{1}{n}\right]+\left[x+\frac{2}{n}\right]+\ldots .+\left[x+\frac{n-1}{n}\right]=[n x]\)

12. Periodic Function

A function is said to be periodic function if its each value is repeated after a definite interval. So a function f(x) will be periodic if a positive real number T exist such that, f(x + T) = f(x) , ∀ x ∈ Domain. Here the least positive value of T is called the period of the function. For example, sin x, cos x, tan x are periodic functions with period 2π, 2π & π respectively.
Note:
(i) If function f (x) has period T then

• f(nx) has period T/n
• f(x/n) has period nT
• f[ax + b) has period \(\frac{T}{|a|}\)

(ii) If the period of f(x) and g(x) are same (T) then the period of af(x) + bg(x) will also be T.

(iii) If the period of f(x) is T₁ and g(x) has T₂, then the period of f(x) ± g(x) will be LCM of T₁ and T₂ provided it satisfies the definition of periodic function.

13. Kinds of mapping

(i) One-one Function or Injection:
A function f: A → B is said to be one-one if different elements of A have different images in B.

(ii) Many-one Function:
A function f: A → B is called many-one, if two or more different elements of A have the same f-image in B.

(iii) Onto Function or Suijection:
A function f: A → B is onto if the each element of B has its pre-image in A.
In other words, range of f = Co-domain off

(iv) Into Function:
A function f: A → B is into if there exist atleast one element in B which is not the f-image of any element in A.
In other words, range of f ≠ co-domain off

(v) One-one onto Function or bijection:
A function f is said to be one-one onto if f is one-one and onto both.

(vi) One-one into Function:
A function is said to be one-one into if f is one-one but not onto.

(vii) Many one-onto Function:
A function f: A → B is many one- onto if f is onto but not one-one.

• f: R → R⁺ ∪ {0}, f(x) = x².
• f: R → [0, ∞), f(x) = |x|

(viii) Many one-into Function:
A function is said to be many one-into if it is neither one-one nor onto.

• f: R → R, f(x) = sin x
• f: R → R, f(x) = |x|

(ix) Identity Function:
Let A be any set and the function f: A → A be defined as f(x) = x , ∀ x ∈ A i.e. if each element of A is mapped by itself then f is called the identity function. It is represented by I_A. If A = {x, y, z} then I_A = {(x, x), (y, y), (z, z)}

14. Composite function

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.
Properties of Composite Function

• If f and g are two functions then for composite of two functions
  fog ≠ gof.
• Composite functions obeys the property of associativity i.e.
  fo(goh) = (fog)oh.
• Composite function of two one-one onto functions if exist, will also be a one-one onto function.

15. Inverse Function

If f: A → B be a one-one onto (bijection) function, then the mapping f^-1: B → A which associates each element b ∈ B with element a ∈ A, such that f(a) = b, is called the inverse function of the function f: A → B
f^-1: B → A, f^-1(b) = a ⇒ f(a) = b
Note: For the existence of inverse function, it should be one-one and onto.

16. Modulus function:

It is given n ∈ N by y = |x| = \(\left\{\begin{array}{ll}\mathrm{x}, & \mathrm{x} \geq 0 \\-\mathrm{x}, & \mathrm{x}<0\end{array}\right.\)
Properties of Modules function:

• |x| ≤ a ⇒ -a ≤ x ≤ a
• |x| ≥ a ⇒ x ≤ – a or x ≥ a
• |x + y| = |x| + |y| x, y ≥ 0 or x ≤ 0, y ≤ 0
• |x – y| = |x| – |y| ⇒ x ≥ 0 and |x| ≥ |y| or x ≤ 0 and y ≤ 0 and |x| ≥ |y|
• |x ± y| ≤ |x| + |y|
• |x ± y| ≥ |x| -|y|

17. Even Extension:

If a function f(x) is defined on the interval [0, a], 0 ≤ x ≤ a ⇒ -a ≤ -x ≤ 0 we define f(x) in the [- a, 0] such that f(x) = f(-x). Let
I(x) = \(\left\{\begin{array}{l}f(x) \quad: x \in[0, a] \\f(-x): x \in[-a, 0]\end{array}\right.\)

18. Odd Extension:

If a function f(x) is defined on the interval [0, a], 0 ≤ x ≤ a ⇒ -a ≤ -x ≤ 0
∴ x ∈ [-a, 0], we define f(x) = – f(-x). Let be the odd extension then
I(x) = \(\left\{\begin{array}{l}f(x), x \in[0, a] \\-f(-x), x \in[-a, 0]\end{array}\right.\)

19. Signum function:

The signum function f is defined as
sgn(x) = \(\left\{\begin{array}{ll}1, & \text { if } x>0 \\0, & \text { if } x=0 \\-1, & \text { if } x<0\end{array}\right.\)

20. Fractional Part function:

It is denoted as f(x) = {x} and defined as

• {x} = f if x = n + f where n e I and 0 ≤ f ≤ 1
• {x} = x – [x]

Keep in mind → For proper fraction 0 < f < 1,

Some very important point

21. If x, y are independent variables then:

• If f(xy) = f(x) + f(y) ⇒ f(x) = k logx
• If f(xy) = f(x). f(y) ⇒ f(x) = xⁿ, n ∈ R
• If f(x + y) = f(x).f(y) ⇒ f(x) = a^kx
• If f(x + y) = f(x) + f(y) ⇒ f(x) = x
• If f(x + y) = f(x) = f(y) ⇒ f(x) = k, here k is constant
• By considering a general nth degree polynomial and writing the expression
  f(x) . f\(\left(\frac{1}{x}\right)\) = f(x) + f\(\left(\frac{1}{x}\right)\) ⇒ f(x) = ± xⁿ + 1

22. Transdental function:

All those function who has infinite terms while expanded are called transdental function. For example, all trigonometrical function, Inverse trigonometrical function, exponential function, logarithmic function etc.

23. Mapping:

(i) One-one or injective mapping or monomorphism:
If f: A → B is one-one mapping, A has m element and B has n element hence the no. of mappings = \(\left\{\begin{array}{l}{ }^{n} P_{m}, n \geq m \\0 \quad, n<m\end{array}\right.\)
(ii) many-one mapping.
(iii) onto mapping or surjective mapping
(iv) into mapping

Four type of fucntion based on mapping classification.

• one-one, onto OR Bijective mapping
• one-one into mapping
• many-one onto mapping
• many-one into mapping

24. Algebra of function:

• (fog) (x) = f [g(x)]
• (fof) (x) = f [f(x)]
• (gog) (x) = g[g(x)]
• (fg) (x) = f(x). g(x)
• (f ± g)(x) = f(x) ± g(x)
• (f/g)(x)= \(\frac{f(x)}{g(x)}\), g(x) ≠ 0
  • Composite functions is not commutative j
  • Let f and g are two functions then if f & g are injective or suijective or bijective then “gof” also injective or surjective or bijective.

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