Domain, Range & Functions

Calculus Worksheet

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 = $\frac{4}{3}\pi r^3$ .

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 $\rightarrow$ B where y = f(x), $x\in A, y\in B$ .

If y = f(x), the we say that y is the image of x under f

and $x$ is the pre-image of y under f.

II. Domain, Range & Co-domain of a Function.

If f: A $\rightarrow$ B 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: A $\rightarrow$ 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 $\rightarrow$ B is called a Real function.

IV. Algebra of Functions:

If f: A $\rightarrow$ R and g: A $\rightarrow$ 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)
$\left ( \frac{f}{g} \right )(x)$ = $\frac{f(x)}{g(x)}$ , g(x) $\neq$ 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 $f(x)=x^11 +3x$ at x = 1 .

Value of f(x) at (x=1)= f(1)= $\left ( 1^{11} \right )$ + 3(1)=1+3=4.

Example 2: If f(x)= $x^2$ and g(x)=x+1 then find (f+g),(f-g) ,(f.g) & $\left ( \frac{f}{g} \right )$ .

(f+g)(x)=f(x)+g(x)= $x^2$ +(x+1)= $x^2$ +1.

(f-g)(x)=f(x)-g(x)= $x^2$ -(x+1)= $x^2$ -1-1.

(f.g)(x)=f(x).g(x)= $x^2$ (x+1)= $x^3$ + $x^2$ .

$\left ( \frac{f}{g} \right )$ (x)= $\frac{f(x)}{g(x)}$ = $\frac{x^2}{(x+1)}$ ,g(x)=(x+1) $\neq$ 0.

Example 3: Find the value of f(5)& 3 f(x) if f(x)= $\frac{x^2-1}{x^2-3x+7}$ .

f(x)= $\frac{x^2-1}{x^2-3x+7}$

f(5)= $\frac{(5)^2-1}{(5)^2-3(5)+7}$ = $\left ( \frac{25-1}{25-15+7} \right )$ = $\frac{24}{17}$

3f(x)=3[f(x)]=3 $\left [ \frac{x^2-1}{x^2-3x+7} \right ]$ = $\left [ \frac{3x^2-3}{x^2-3x+7} \right ]$

Check Point

Find the value of at f(x) = $x^4$ +20x at x=2.

Find the value of at f(x) = $x^4$ +2 $x^3$ -6x-9 at x = -1.

Find the value of f(2) if f(x) = $\frac{x-1}{x^2+7x+5}$ .

If f(x)=(x+2)&g(x)=(x-2) then find(f+g) ,(f-g) ,(f.g) & $\left ( \frac{f}{g} \right )$ .
If f(x)= $x^2$ +2x+3& g(x)= ( $x^2$ -3) then find (f+g), (f-g), (f.g)& $\left ( \frac{f}{g} \right )$ .

Answer Key

56
-4
f(2)= $\frac{1}{23}$

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