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) = a_{0} + a_{1}x + a_{2}x^{2} + …..a_{n}x^{n}, where n is a nonnegative integer and a_{0, }a_{1, }a_{2 … }a_{n }∈_{ }R_{.}
E.g., f (x) = x^{2} 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 nonnegative 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) = 2x^{2} + 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) = 2x^{2} + 1















Solution:
The completed table is as follows:
x

−3

−2

−1

0

1

2

3

y = f (x) = 2x^{2} + 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 − x^{2} ≠ 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) = [log_{10} (5 − x)]^{− 1} and
Then f(x) = g(x) + h(x)
Also Domain (f) = Domain (g) ∩ Domain (h)
Now, g(x) is defined if
Log_{10} (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)