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

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)= $\frac{1}{23}$

5.

Personalized Online Tutoring

eTutorWorld offers affordable one-on-one live tutoring over the web for Grades K-12, Test Prep help for Standardized tests like SCAT, CogAT, MAP, SSAT, SAT, ACT, ISEE and AP. You may schedule online tutoring lessons at your personal scheduled times, all with a Money-Back Guarantee. The first one-on-one online tutoring lesson is always FREE, no purchase obligation, no credit card required.

For answers/solutions to any question or to learn concepts, take a FREE TRIAL Session.

Schedule a Free Session

No credit card required, no obligation to purchase.
Just schedule a FREE Sessions to meet a tutor and get help on any topic you want!

Pricing for Online Tutoring

Tutoring Package	Validity	Grade (1-12), College
5 sessions	1 Month	$129
1 session	1 Month	$26
10 sessions	3 months	$249
15 sessions	3 months	$369
20 sessions	4 months	$469
50 sessions	6 months	$1099
100 sessions	12 months	$2099