Types of Functions

In the last article, we have studied the concept of the function and the terminology associated with it. In this article, we shall study different types of functions.

Real Function:

A function whose domain and co-domain are the set or subset of real numbers R, then the function is called a real function.
Thus if ƒ: R → R then ƒ is a real function.

Example:

Consider function y = ƒ(x) = x² + 3x + 2

For every x ∈ R, y = ƒ(x) = x² + 3x + 2 ∈ R,

Thus the function y = ƒ(x) = x² + 3x + 2 is a real function.

Constant Function:

If a real function ƒ is defined as ƒ(x) = k, k is constant for all x ∈ R Then is called a constant function.

Examples: 

ƒ(x) = 3, ƒ(x) = – 4 etc.

Note:

• The domain for the constant function is a set of real number R, i.e. D_ƒ = R, while its range is {k}
• The range contains only one element. i.e. R_ƒ = {k}
• Constant function is many-one function
• The graph for a constant function is as follows.

Zero Function:

For constant function k = 0 then the function is called zero function.

Example:

ƒ(x) = 0, g(x) = 0 etc

Note:

• The domain for zero function is a set of real number R i.e. D_ƒ = R, while its range is {0}.
• The range contains only one element i.e. zero. i.e. R_ƒ = {0}
• Zero function is many-one function.
• The graph for the zero function is as follows.

The graph is x-axis

Identity Function:

If real function ƒ is defined as ƒ(x) = x, for all x ∈ R Then ƒ is called identity function.

Example:

y = x

Note:

The domain and range for identity function is a set of real number R i.e. D_ƒ = R.

The graph for the zero function is as follows.

Absolute value function:

A function ƒ is defined by ƒ(x) = |x|, Where

is called an absolute value function.

Note:

The domain for absolute value function is a set of real number R i.e. D_ƒ = R.

The graph for the absolute value function is as follows.

Signum function:

A function is defined by

is called signum function.

Note:

• The domain for signum function is a set of real number R i.e. D_ƒ = R.
• The range of signum function contains three elements only. = {-1, 0, 1}
• The graph for the signum function is as follows.

Greatest integer function:

The greatest integer function ƒ is defined as [x], the greatest integer ≤ x, for each x ∈ R. Thus, [x] = x if x is integer and [x] = an integer immediately on the left side of x if x is not an integer.

Examples:

[5] = 5, [-6.9] = -7, [0] = 0, [2.3] = 2, [17/3] = 5

Note:

• The domain for greatest integer function is a set of real number R i.e. D_ƒ = R.
• The range of greatest integer function is a set of integers. R_ƒ =  I
• The graph for the greatest integer function is as follows.

Fractional part function:

A functionƒ defined by ƒ(x) = x – [x], is called fractional part function.

Examples: 

(3.9) =3.9 -3 = 0.9 and (-6.9) = -6.9 – (-7) = 0.1

Note:

• The domain for fractional part function is a set of real number R i.e. D_ƒ =  R.
• Range of fractional part function is R_ƒ =  [0, 1) i.e. 0 ≤ f(x) < 0
• The graph for the fractional part function is as follows.

Linear function:

A function defined by ƒ(x) = mx + c, where m, c ∈ R and m ≠ 0 is called a linear function.

Example:

ƒ(x) = y = 3x + 5

Note:

• The domain for a linear function is a set of real number R i.e. D_ƒ =  R.
• The range of a linear function is a set of real numbers. R_ƒ = R
• The graph of a linear function is a straight line.
• If c = 0 then the graph passes through the origin.

Polynomial Function:

If real function ƒ is defined as ƒ(x) = a₀ + a₁x + a₂x² + a₃x³ + ……… +a_nxⁿ. Where a₀, a₁, a₂, a₃, …,a_n ∈ R and n is a whole number. Then ƒ is called as a polynomial function.

Example :

ƒ(x) = x² + 3x + 2

Note:

• The domain and range for a polynomial function is a set of real number R. Thus, D_ƒ = R. and R_ƒ = = R

Reciprocal function:

A function ƒ defined by ƒ(x) = 1/x, Where x ∈ R and x ≠ 0. is called reciprocal function.

Note:

• The domain for reciprocal function is a set of real number R except x ≠ 0 i.e. D_ƒ = R – {0}.
• The graph for the reciprocal function is as follows.

Exponential function:

A function ƒ defined by ƒ(x) = e^x is called exponential function.

Note:

• The domain for the exponential function is a set of real number R i.e. D_ƒ = R
• The graph for the exponential function is as follows.

Logarithmic function:

A function ƒ defined by ƒ(x) =log x, x > 0 is called logarithmic function.

Note:

• The domain for the logarithmic function is a set = {x| x ∈ R and x > 0}
• The graph for the logarithmic function is as follows.

Trigonometric functions:

Graphs of Trigonometric Functions:

Inverse trigonometric functions:

Some Important Results of Inverse Functions:

SET – I

1. sin(sin^-1x) = x, for |x|<1
2. sin^-1(sin x) = x, for |x| ≤ π/2
3. cos(cos^-1x) = x, for |x|<1
4. cos^-1(cos x) = x, for x = [0, π]
5. tan(tan^-1x) = x, ∀ x ∈ R
6. tan^-1(tan x) = x, for x ∈ (- π/2, π/2)
7. cot(cot^-1x) = x, x ∈ R
8. cot^-1(cot x) = x, for x ∈ (0, π)
9. sec(sec^-1x) = x, |x| ≥ 1
10. sec^-1(sec x) = x, for x ∈ [0, – π/2) ∪ (π/2, π]
11. cosec(cosec^-1x) = x, |x| ≥ 1
12. cosec^-1(cosec x) = x, for x ∈ [- π/2, 0) ∪ (0, π/2]

SET- II

1. cosec^-1x = sin^-1(1/x)
2. sec^-1x = cos^-1(1/x)
3. cot^-1x = tan^-1(1/x)

SET – III

1. sin^-1x + cos^-1x = π/2
2. tan^-1x + cot^-1x = π/2
3. sec^-1x + cosec^-1x = π/2

SET – IV

SET – V

Rational functions:

A function ƒ of the form p(x)/q(x) = 0, q(x) ≠ 0 is called rational function. Its domain is R except for q(x) ≠ 0