Here you will learn types of functions in maths i.e polynomial function, logarithmic function etc and their domain and range.
Let’s begin –
Types of Functions in Maths
(a) Polynomial function
If a function is defined by f(x) = \(a_0x^n\) + \(a_1x^{n-1}\) + \(a_2x^{n-2}\) + ….. + \(a_{n-1}x\) + \(a_n\) where n is a non negative integer and \(a_0\), \(a_1\), \(a_2\), ….. , \(a_n\) are real numbers and \(a_0\) \(\ne\) 0, then f is called a polynomial function of degree n. If n is odd, then polynomial is of odd degree, if n is even, then the polynomial is of even degree.
Note :
(i) Range of odd degree polynomial is always R.
(ii) Range of even degree polynomial is never R.
(iii) A Polynomial of degree one with no constant term is called an odd linear function. i.e. f(x) = ax,
a \(\ne\) 0
(iv) f(x) = ax + b, a \(\ne\) 0 is a linear polynomial.
(v) f(x) = c, is a non linear polynomial(its degree is zero).
(vi) f(x) = 0, is a polynomial but its degree is not defined.
(vii) There are two polynomial functions, satisfying the relation;
f(x).f(1/x) = f(x) + f(1/x). They are :
(a) f(x) = \(x^n\) + 1 and (b) f(x) = 1 – \(x^n\), where n is a positive integer.
(b) Algebraic Function
A function f is called an algebraic function if it can be constructed using algebraic operations(such as addition, subtraction, multiplication, division, and taking radicals) within polynomials.
Note :
(i) All polynomial functions are algebraic but not the converse.
(ii) A function that is not algebraic is called Transcendental function.
(c) Rational Function
A rational function is a function of the form y = f(x) = \(g(x)\over h(x)\), where g(x) & h(x) are polynomials & h(x) \(\ne\) 0,
Domain : R – {x | h(x) = 0}
Any algebraic function is automatically an algebraic function.
(d) Trigonometric Functions
(i) Sine Function
f(x) = sinx
Domain : R
Range : [-1, 1], period 2\(\pi\)
(ii) Cosine Function
f(x) = cosx
Domain : R
Range : [-1, 1], period 2\(\pi\)
(iii) Tangent Function
f(x) = tanx
Domain : R – {x | x = \({(2n+1)\pi\over 2}\), n \(\in\) I }
Range : R, period \(\pi\)
(iv) Cosecant Function
f(x) = cosecx
Domain : R – { x | x = n\(\pi\), n \(\in\) I }
Range : R – (-1, 1), period 2\(\pi\)
(v) Secant Function
f(x) = secx
Domain : R – { x | x = (2n+1)\(\pi\)/2, n \(\in\) I }
Range : R – (-1, 1), period 2\(\pi\)
(vi) Cotangent Function
f(x) = cotx
Domain : R – { x | x = n\(\pi\), n \(\in\) I }
Range : R, period \(\pi\)
(e) Exponential and Logarithmic function
A function f(x) = \(a^x\)(a > 0), a \(\ne\) 1, x \(\in\) R is called an exponential function. The inverse of the exponential function is called the logarithmic function, i.e. g(x) = \(log_ax\).
Note that f(x) & g(x) are inverse of each other.
Domain of \(a^x\) is R Range \(R^+\)
Domain of \(log_ax\) is \(R^+\) Range R
(f) Absolute Value Function
A function y = f(x) = |x| is called the absolute value function or modulus function. It is defined as :
y = |x| = [x if x \(\le\) 0 -x if x < 0]
For f(x) = |x|, domain is R and range is [0,\(\infty\)]
For f(x) = \(1\over{|x|}\), domain is R – {0} and range is \(R^+\)
(g) Signum Function
A function y = f(x) = Sgn(x) is defined as follows :
y = f(x) = [ 1 for x > 0 0 for x = 0 -1 for x < 0 ]
Domain : R
Range : [-1, 0, 1]
(h) Greatest integer or step up function
The function y = f(x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x. Note that for :
| x | [x] |
| [-2,-1] | -2 |
| [-1,0] | -1 |
| [0,1] | 0 |
| [1,2] | 1 |
Domain : R
Range : I
(i) Fractional part function
It is defined as : g(x) = {x} = x – [x] e.g. the fractional part of the number 2.1 is 2.1 – 2 = 0.1 and the fractional part of -3.7 is 0.3. The period of this function is 1.
(j) Identity function
The function f : A \(\rightarrow\) A defined by f(x) = x \(\forall\) x \(\in\) A is called the identity of A and is denoted by \(I_A\). It is easy to observe that identity function defined on R is bijection.
(j) Constant function
The function f : A \(\rightarrow\) B is said to be a constant function if every element of A has the same f image in B. Thus f : A \(\rightarrow\) B ; f(x) = c, \(\forall\) x \(\in\) A, c \(\in\) B is a constant function. Note that the range of constant function is singleton.
Hope you learnt types of functions in maths and to learn more practice more questions and get ahead in competition. Good Luck!