Functions are relations where each input has a particular output. This lesson covers the concepts of functions in mathematics and the different types of functions using various examples for better understanding. Show
Contents Related to Functions
JEE Main 2021 Maths LIVE Paper Solutions 24-Feb Shift-1 Memory-BasedWhat are Functions in Mathematics?A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets; mapping from A to B will be a function only when every element in set A has one end, only one image in set B. Example: Another definition of functions is that it is a relation “f” in which each element of set “A” is mapped with only one element belonging to set “B”. Also in a function, there can’t be two pairs with the same first element. A Condition for a Function:Set A and Set B should be non-empty. In a function, a particular input is given to get a particular output. So, A function f: A->B denotes that f is a function from A to B, where A is a domain and B is a co-domain.
The unique element b to which f relates a, is denoted by f(a) and is called f of a, or the value of f at a, or the image of a under f.
A real-valued function has either P or any one of its subsets as its range. Further, if its domain is also either P or a subset of P, it is called a real function. Vertical Line Test: Vertical line test is used to determine whether a curve is a function or not. If any curve cuts a vertical line at more than one points then the curve is not a function. Representation of FunctionsFunctions are generally represented as f(x). Let , f(x) = x3. It is said as f of x is equal to x cube. Functions can also be represented by g(), t(),… etc. Steps for Solving FunctionsQuestion: Find the output of the function g(t) = 6t2 + 5 at (i) t = 0 (ii) t = 2 Solution: The given function is g(t) = 6t2 + 5 (i) At t = 0, g(0) = 6(0)2 + 5 = 5 (ii) At t = 2, g(2) = 6(2)2 + 5 = 29 Types of FunctionsThere are various types of functions in mathematics which are explained below in detail. The different function types covered here are:
Practice: Find the missing equations from the above graphs. Functions – Video LessonsFunctions and Types of FunctionsNumber of FunctionsEven and Odd FunctionsComposite and Periodic FunctionsOne – one function (Injective function)If each element in the domain of a function has a distinct image in the co-domain, the function is said to be one – one function. For examples f; R R given by f(x) = 3x + 5 is one – one. Many – one functionOn the other hand, if there are at least two elements in the domain whose images are same, the function is known as many to one. For example f : R R given by f(x) = x2 + 1 is many one. Onto – function (Surjective Function)A function is called an onto function if each element in the co-domain has at least one pre – image in the domain. Into – functionIf there exists at least one element in the co-domain which is not an image of any element in the domain then the function will be Into function. (Q) Let A = {x : 1 < x < 1} = B be a mapping f : A B, find the nature of the given function (P). F(x) = |x| f (x) = |1|Solution for x = 1 & -1 Hence, it is many one the Range of f(x) from [-1, 1] is [0, 1], which is not equal to the co-domain. Hence, it is into function. Lets say we have function, \(\begin{array}{l}f(x)=\left\{\begin{matrix} x^2 & ; & x\geq 0\\ -x^2 & ; & x<0 \end{matrix}\right.\end{array} \) For different values of Input, we have different output hence it is one – one function also it manage is equal to its co-domain hence it is onto also. Polynomial functionA real-valued function f : P → P defined by \(\begin{array}{l}y = f(a) = h_{0}+h_{1}a+…..+h_{n}a^{n}\end{array} \) , where n ∈ N and h0 + h1 + … + hn ∈ P, for each a ∈ P, is called polynomial function.
So, a polynomial function can be expressed as : \(\begin{array}{l}f(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{1}x^{1}+a_{0}\end{array} \) The highest power in the expression is known as the degree of the polynomial function. The different types of polynomial functions based on the degree are:
Linear FunctionAll functions in the form of ax + b where a, b ∈ R & a ≠ 0 are called linear functions. The graph will be a straight line. In other words, a linear polynomial function is a first-degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c. For example, f(x) = 2x + 1 at x = 1 f(1) = 2.1 + 1 = 3 f(1) = 3 Another example of linear function is y = x + 3
Identical FunctionTwo functions f and g are said to be identical if (a) The domain of f = domain of g (b) The range of f = the Range of g (c) f(x) = g(x) ∀ x ∈ Df & Dg For example f(x) = x \(\begin{array}{l}g(x) = \frac{1}{1/x}\end{array} \) Solution: f(x) = x is defined for all x But \(\begin{array}{l}g(x) = \frac{1}{1/x}\end{array} \) is not defined of x = 0Hence it is identical for x ∈ R – {0} Quadratic FunctionAll functions in the form of y = ax2 + bx + c where a, b, c ∈ R, a ≠ 0 will be known as Quadratic function. The graph will be parabolic. \(\begin{array}{l}\text{At}\ x=\frac{-b \pm \sqrt{D}}{2}\end{array} \) , we will get its maximum on minimum value depends on the leading coefficient and that value will be -D/4a (where D = Discriminant)In simpler terms, A Quadratic polynomial function is a second degree polynomial, and it can be expressed as; F(x) = ax2 + bx + c, and a is not equal to zero. Where a, b, c are constant, and x is a variable. Example, f(x) = 2x2 + x – 1 at x = 2 If x = 2, f(2) = 2.22 + 2 – 1 = 9 For Example: y = x2
Read More: Quadratic Function Formula Rational FunctionThese are the real functions of the type \(\begin{array}{l}\frac{f(a)}{g(a)}\end{array} \) where f (a) and g (a) are polynomial functions of a defined in a domain, where g(a) ≠ 0.
Algebraic FunctionsAn algebraic equation is known as a function that consists of a finite number of terms involving powers and roots of independent variable x and fundamental operations such as addition, subtraction, multiplication, and division. For Example, \(\begin{array}{l}f(x)=5x^{3}-2x^{2}+3x+6\end{array} \) ,\(\begin{array}{l}g(x)=\frac{\sqrt{3x+4}}{(x-1)^{2}}\end{array} \) .Cubic FunctionA cubic polynomial function is a polynomial of degree three and can be expressed as; F(x) = ax3 + bx2 + cx + d and a is not equal to zero. In other words, any function in the form of f(x) = ax3 + bx2 + cx + d, where a, b, c, d ∈ R & a ≠ 0 For example: y = x3 Domain ∈ R Range ∈ R Modulus FunctionThe real function f : P → P defined by f (a) = |a| = a when a ≥ 0. and f(a) = -a when a < 0 ∀ a ∈ P is called the modulus function.
\(\begin{array}{l}y=|x|=\left\{\begin{matrix} x & x\geq 0\\ -x & x<0 \end{matrix}\right.\end{array} \) Domain: R Range: [0, ∞) Signum FunctionThe real function f : P → P is defined by \(\begin{array}{l}\left\{\begin{matrix}\frac{\left | f(a) \right |}{f(a)}, a\neq 0 \\ 0, a=0 \end{matrix}\right. = \left\{\begin{matrix} 1,if x>0\\ 0, if x=0\\ -1, if x<0\end{matrix}\right.\end{array} \) is called the signum function or sign function. (gives the sign of real number)
For example: signum (100) = 1 signum (log 1) = 0 signum (x21) =1 Greatest Integer FunctionThe real function f : P → P defined by f (a) = [a], a ∈ P assumes the value of the greatest integer less than or equal to a, is called the greatest integer function.
The greatest integer function always gives integral output. The Greatest integral value that has been taken by the input will be the output. For example: [4.5] = 4 [6.99] = 6[1.2] = 2Domain ∈ R Range ∈ Integers Fractional Part Function{x} = x – [x] It always gives fractional value as output. For example:- {4.5} = 4.5 – [4.5] = 4.5 – 4 = 0.5 {6.99} = 6.99 – [6.99] = 6.99 – 6 = 0.99 {7} = 7 – [7] = 7 –7 = 0 Even and Odd FunctionIf f(x) = f(-x) then the function will be even function & f(x) = -f(-x) then the function will be odd function Example 1: f(x) = x2sinx f(-x) = -x2sinx Here, f(x) = -f(-x) It is an odd function. Example 2: \(\begin{array}{l}f(x)={{x}^{2}}\end{array} \) and \(\begin{array}{l}f(-x)={{x}^{2}}\end{array} \) f(x) = f(-x) It is an even function. Periodic FunctionA function is said to be a periodic function if a positive real number T exists, such that f(u – t) = f(x) for all x ε Domain. For example f(x) = sin x f(x + 2π) = sin (x + 2π) = sin x fundamental then period of sin x is 2π Composite FunctionLet A, B, C be three non-empty sets Let f: A → B & g : G → C be two functions, then gof : A → C. This function is called the composition of f and g given gof (x) = g(f(x)). For example f(x) = x2 & g(x) = 2x f(g(x)) = f(2x) = (2x)2 = 4x2 g(f(x)) = g(x2) = 2x2 Constant FunctionThe function f : P → P is defined by b = f (x) = D, a ∈ P, where D is a constant ∈ P, is a constant function.
In simple words, the polynomial of 0th degree where f(x) = f(0) = a0 = c. Regardless of the input, the output always results in a constant value. The graph for this is a horizontal line. Identity FunctionP= set of real numbers The function f : P → P defined by b = f (a) = a for each a ∈ P is called the identity function.
Functions VideoDomain, Range, Period of FunctionsFunctions and RelationsRelations and Functions QuestionsOne-One and Onto FunctionsFrequently Asked QuestionsA relation f from set A to set B is called a function if every element of set A has one and only one image in set B. The domain of a function is the set of all possible inputs for a function. The range of a function is the set of all possible output values. Constant function is a function whose output is the same for every input value. For example, f(x) = 3. Here for every value of x, output will be 3. |