# please do answerfind the domain and range ofa) identity functionb)constant functionc)polynomial functiond)rational functione)modulus functionf)signum functiong) greatest integer function

It seems that you have problem in understanding the concept of finding the domain and range of functions. Please visit www.meritnation.com (chapter - Relations and Functions) Grade XI to understand your problem. If you have problem in finding range and domain of a particular function, then send us the function along with the question and we would be happy to help you.

• -4

hello nihla .... we request you to read the chapter again in textbook and revision notes and study material of the chapter....these all things are given over there.... if you have any further doubt please do post your queries........

• -1

only some are given..nt evrythin ! -_-

• 5

Some Common Real Valued Functions

• A real function f: RR defined by y = f (x) = x is called an identity function. The domain as well as the range of this function is R. Its graph is a straight line passing through the origin. • A function f: RR defined by y = f (x) = c, xR,where c is a constant is called a constant function. The domain of f isR and its range is {c}. • A function f: RR is said to be a polynomial functionif for each x in R, y = f (x) = a0 + a1x + a2x2 + …..nxn, where n is a non-negative integer and a0, a1, a2 … n R.
E.g., f (x) = x2 is a polynomial function and its graph is given below. • Rational functions are of the form , where f (x) and g (x) are polynomial functions of x defined in a domain, where g (x) ≠ 0. E.g., is a rational function and its graph is given below. • The function f: RR defined by y = f (x) = for each x R is called modulus function.
For non-negative values of x, f (x) = x and for negative values of x, f (x) = −x. Modulus function can also be defined as The graph of the modulus function is given below. • The function f: RR defined by is called the signum function. The domain of the signum function is R and the range is the set {−1, 0, 1}. The graph of the signum function is given below. • The function f: RR defined by f (x) = [x], xR assumes the value of the greatest integer, less than or equal to x. Such a function is called the greatest integer function. E.g., [x] = 3 for 3 ≤ x < 4. The graph of the function is given below. Solved Examples

Example 1:

Define the function f: R → R by y = f (x) = 2x2 + 1, x ∈ R. Complete the given table by using this function. What is the domain and range of this function? Draw the graph of f.

 x −3 −2 −1 0 1 2 3 y = f (x) = 2x2 + 1 - - - - - - -

Solution:

The completed table is as follows:

 x −3 −2 −1 0 1 2 3 y = f (x) = 2x2 + 1 19 9 3 1 3 9 19

Domain of f = {x: x R}

Range of f = {x: x ≥ 1, x R}

The graph of f is given below. Example 2:

Find the domain and range of f(x) = .

Solution:

We have f(x) = Now, f(x) is defined for all xR.

Domain of f = R

Now, let us find the range of f.

We know that for all xR.

Hence, we have Hence, range of f = Example :

Find the range of function f(x) given by .

For f(x) to be real

3 − x2 ≠ 0 ∴ Domain Let  Since x is real,  ∴ Range of Example :

Find the range of function . Now, f(x) is defined if  ∴ Domain of f (x) = [− 3, 3]

Let  x will be real if  Also, for all x ∈ [− 3, 3]

y ∈ [0. 3] for all x ∈ [− 3, 3]

∴ Range of f(x) = [0, 3]

Example :

Find the range of the function f(x) = .

− 1 ≤ sin 5x ≤ 1 for all x ∈ R

⇒ − 1 ≤ − sin 5x ≤ 1 for all x ∈ R … (1)

⇒ 3 ≤ 4 − sin 5x ≤ 5 for all x ∈ R

∴ 4 − sin 5x ≠ 0 for all x ∈ R

f(x) = is defined for all x ∈ R

Thus, the domain of f(x) = R

From equation (1): Hence, range of Example :

What is the domain of definition of the function f(x) given by ?

Let g(x) = [log10 (5 − x)]− 1 and Then f(x) = g(x) + h(x)

Also Domain (f) = Domain (g) ∩ Domain (h)

Now, g(x) is defined if

Log10 (5 − x) is defined and ≠ 0 Domain (g) = (−∞, 4) ∪ (4, 5)

And h(x) is defined if x + 3 ≥ 0

x ∈ [−3, ∞]

∴ Domain (h) = [−3, ∞]

Hence, Domain (f) = (−∞, 4) ∪ (4, 5) ∩ [−3, ∞)

= [−3, 4) ∪ (4, 5)

• 2
What are you looking for?