# Functions

**Have you ever inflated a spherical balloon?**

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.**

**I. 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*: *A**B* 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*: *A**B* is a function from set *A* to set *B*, then **set A **is called the

**Domain**of

*f*and

**set**is called the

*B***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*: *A**B* 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:**

If* f*: *A**R* and* g*: *A**R* are any two functions then we have:

- (f+g)(x)=f(x)+g(x)
- (f-g)(x)=f(x)-g(x)
- (f.g)(x)=f(x).g(x)
- =, 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*.

**Examples*** *

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) & .

(f+g)(x)=f(x)+g(x)=+(x+1)=+1.

(f-g)(x)=f(x)-g(x)=-(x+1)=-1-1.

(f.g)(x)=f(x).g(x)=(x+1)=+.

(x)==,g(x)=(x+1)0.

** Example 3:** Find the value of f(5)& 3 f(x) if f(x)= .

f(x)=

f(5)===

3f(x)=3[f(x)]=3=

- 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)& .

#### Answer Key

- 56
- -4
- f(2)=

5.

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