Concept Of Function:
The term function was recognized by a German Mathematician Leibniz (1646 - 1716) to describe the dependence of one quantity on another. The following examples illustrates how this term is used:
- The area "A" of a square depends on one of its sides "x" by the formula A = x2 , so we say that A is a function of x.
- The volume "V" of a sphere depends on its radius "r" by the formula V = (4/3)πr³ , so we say that V is a function of r.
A Function f from a set X to a set Y is a rule or a correspondence that assigns to each element x in X a unique element y in Y. The set X is called the domain of f. The set of corresponding elements y in Y is called the range of f.
Unless stated to the contrary, we shall assume hereafter that the sets X and Y consist of real numbers.
Important Note:
Functions are often denoted by the letters such as f, g, h, F, G, H and so on.
y = f(x)
The variable x is called the independent variable of f, and the variable y is called the dependent variable of f.
Types Of Functions:
Some important types of functions are given below:
1. Algebraic Functions:
- Polynomial Function
- Linear Function
- Identity Function
- Constant Function
- Rational Function
3. Inverse Trigonometric Functions
4. Exponential Function
5. Logarithmic Function
6. Hyperbolic Functions
7. Inverse Hyperbolic Functions
8. Explicit Function
9. Implicit Function
10. Even Function
11. Odd Function
Limit Of A Function and Theorems On Limits:
Let a function f(x) be defined in an open interval near the number "a" (need not at a).
If, as x approaches "a" from both left and right side of "a", f(x) approaches a specific number "L" then "L" is called the limit of f(x) as x approaches a.
Symbollically it is written as:
Lim x→a f(x) = L read as "limit of f(x), as x→a, is L"
- Parametric Functions
10. Even Function
11. Odd Function
Limit Of A Function and Theorems On Limits:
Let a function f(x) be defined in an open interval near the number "a" (need not at a).
If, as x approaches "a" from both left and right side of "a", f(x) approaches a specific number "L" then "L" is called the limit of f(x) as x approaches a.
Symbollically it is written as:
Lim x→a f(x) = L read as "limit of f(x), as x→a, is L"
Theorems On Limits Of Functions:
Theorem 1: The limit of the sum of two functions is equal to the sum of their limits.
Theorem 2: The limit of the difference of two functions is equal to the difference of their limits.
Theorem 3: If k is any real number then
Lim x→a [k f(x)] = k Lim x→a f(x) = k L
Theorem 4: The limit of the product of the functions is equal to the product of their limits.
Theorem 5: The limit of the quotient of the functions is equal to the quotient of their limits provided the limit of the denominator is non-zero.
Theorem 6: Limit of [f(x)]n, where n is an integer.
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