What do we observe? As we start inflating the balloon its radius starts increasing and consequently its volume also starts increasing.
Volume of the spherical balloon is given by V = .
Here Volume V (dependent variable) is a function of radius r (independent variable).
Hence, we introduce the concept of Function.
If A & B are any two nonempty sets. Then a function f from A to B is a rule or correspondence that assigns to each element of set A, one and only one element of B.
A function from A to B is denoted by f: AB where y = f(x), .
If y = f(x), the we say that y is the image of x under f
and is the pre-image of y under f.
II. Domain, Range & Co-domain of a Function.
If f: AB is a function from set A to set B, then set A is called the Domain of f and set B is called the co-domain of f. The set of all images of the elements of set A is called the Range of f.
III. Real valued function: A function f: AB is called a real valued function, if its co-domain B is a subset of the set of real numbers.
Real function: If A & B both are subsets of real numbers, then f: A B is called a Real function.
IV. Algebra of Functions:
- =, g(x) 0.
- (cf)(x)=cf(x), where ‘c’ is a any real number
Note: The value of f(x) at x = a is denoted by f(a) and it is obtained by replacing, that is, substituting x with a.
Now let’s consider some examples on functions.
Example 1: Find the value of at at x = 1 .
Value of f(x) at (x=1)= f(1)=+ 3(1)=1+3=4.
Example 2: If f(x)= and g(x)=x+1 then find (f+g),(f-g) ,(f.g) & .
Example 3: Find the value of f(5)& 3 f(x) if f(x)= .
- Find the value of at f(x) = +20x at x=2.
- Find the value of at f(x) = +2-6x-9 at x = -1.
- Find the value of f(2) if f(x) = .
- If f(x)=(x+2)&g(x)=(x-2) then find(f+g) ,(f-g) ,(f.g) & .
- If f(x)=+2x+3& g(x)= ( -3) then find (f+g), (f-g), (f.g)& .
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