please do answer

find the domain and range of

a) identity function

b)constant function

c)polynomial function

d)rational function

e)modulus function

f)signum function

g) greatest integer function

Some Common Real Valued Functions

• A real function f: R → R 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: R → R defined by y = f (x) = c, x ∈ R,where c is a constant is called a constant function. The domain of f isR and its range is {c}.
  

• A function f: R → R is said to be a polynomial functionif for each x in R, y = f (x) = a₀ + a₁x + a₂x² + …..a­_nxⁿ, where n is a non-negative integer and a_0, a_1, a_{2 …} a­_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: R → R 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: R → R 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: R → R defined by f (x) = [x], x ∈ R 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 x ∈ R.

Domain of f = R

Now, let us find the range of f.

We know that for all x ∈ R.

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)