Functions
A function is a relation that maps each element x of a set A with one and only one element y of another set B. In other words, it is a relation between a set of inputs and a set of outputs in which each input is related with a unique output. A function is a rule that relates an input to exactly one output.
It is a special type of relation. A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B and no two distinct elements of B have the same mapped first element. A and B are the non-empty sets. The whole set A is the domain and the whole set B is codomain.
Representation
A function f: X →Y is represented as f(x) = y, where, (x, y) ∈ f and x ∈ X and y ∈ Y.
For any function f, the notation f(x) is read as “f of x” and represents the value of y when x is replaced by the number or expression inside the parenthesis. The element y is the image of x under f and x is the pre-image of y under f.
Every element of the set has an image which is unique and distinct. If we notice around, we can find many examples of functions.
If we lift our hand upward, it is a function. Waving our hand freely, it is a function. A walk in a circular track, yes it is a type of function. Now you can think of other examples too! A graph can represent a function. The graph is the set of all pairs of the Cartesian product.
Does this mean that every curve in the world defines a function? No, not every curve drawn is a function. How to find it? Vertical line test. If any curve intercepts a vertical line at more than one point, it is a curve only not a function.
Solved Example for You
Problem: Which of the following is a function?
1.
2.
3.
Solution: Figure 3 is an example of function since every element of A is mapped to a unique element of B and no two distinct elements of B have the same pre-image in A.
Types of Functions
One to One Function
A function f: A → B is One to One if for each element of A there is a distinct element of B. It is also known as Injective. Consider if a1 ∈ A and a2 ∈ B, f is defined as f: A → B such that f (a1) = f (a2)
Many to One Function
It is a function which maps two or more elements of A to the same element of set B. Two or more elements of A have the same image in B.
Onto Function
If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function. Onto is also referred as Surjective Function.
One – One and Onto Function
A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function. In other words, the function f associates each element of A with a distinct element of B and every element of B has a pre-image in A.
Other Types of Functions
A function is uniquely represented by its graph which is nothing but a set of all pairs of x and f(x) as coordinates. Let us get ready to know more about the types of functions and their graphs.
Identity Function
Let R be the set of real numbers. If the function f: R→R is defined as f(x) = y = x, for x ∈ R, then the function is known as Identity function. The domain and the range being R. The graph is always a straight line and passes through the origin.
Constant Function
If the function f: R→R is defined as f(x) = y = c, for x ∈ R and c is a constant in R, then such function is known as Constant function. The domain of the function f is R and its range is a constant, c. Plotting a graph, we find a straight line parallel to the x-axis.
Polynomial Function
A polynomial function is defined by y =a0 + a1x + a2x2 + … + anxn, where n is a non-negative integer and a0, a1, a2,…, n ∈ R. The highest power in the expression is the degree of the polynomial function. Polynomial functions are further classified based on their degrees:
- Constant Function: If the degree is zero, the polynomial function is a constant function (explained above).
- Linear Function: The polynomial function with degree one. Such as y = x + 1 or y = x or y = 2x – 5 etc. Taking into consideration, y = x – 6. The domain and the range are R. The graph is always a straight line.
Quadratic Function: If the degree of the polynomial function is two, then it is a quadratic function. It is expressed as f(x) = ax2 + bx + c, where a ≠ 0 and a, b, c are constant & x is a variable. The domain and the range are R. The graphical representation of a quadratic function say, f(x) = x2 – 4 is
- Cubic Function: A cubic polynomial function is a polynomial of degree three and can be denoted by f(x) = ax3 + bx2 + cx +d, where a ≠ 0 and a, b, c, and d are constant & x is a variable. Graph for f(x) = y = x3 – 5. The domain and the range are R.
Rational Function
A rational function is any function which can be represented by a rational fraction say, f(x)/g(x) in which numerator, f(x) and denominator, g(x) are polynomial functions of x, where g(x) ≠ 0. Let a function f: R → R is defined say, f(x) = 1/(x + 2.5). The domain and the range are R. The Graphical representation shows asymptotes, the curves which seem to touch the axes-lines.
Modulus Function
The absolute value of any number, c is represented in the form of |c|. If any function f: R→ R is defined by f(x) = |x|, it is known as Modulus Function. For each non-negative value of x, f(x) = x and for each negative value of x, f(x) = -x, i.e.,
f(x) = {x, if x ≥ 0; – x, if x < 0.
Its graph is given as, where the domain and the range are R.
Signum Function
A function f: R→ R defined by
f(x) = { 1, if x > 0; 0, if x = 0; -1, if x < 0
Signum or the sign function extracts the sign of the real number and is also known as step function.
Greatest Integer Function
If a function f: R→ R is defined by f(x) = [x], x ∈ X. It round-off to the real number to the integer less than the number. Suppose, the given interval is in the form of (k, k+1), the value of greatest integer function is k which is an integer. For example: [-21] = 21, [5.12] = 5. The graphical representation is
Solved Example for You
Question: Which of the following is a function?
Solution: Figure (iii) is an example of a function. Since the given function maps every element of A with that of B. In figure (ii), the given function maps one element of A with two elements of B (one to many). Figure (i) is a violation of the definition of the function. The given function does not map every element of A.
Composite Functions
Suppose f is a function which maps A to B. And there is another function g which maps B to C. Can we map A to C? The mapping of elements of A to C is the basic concept of Composition of functions. When two functions combine in a way that the output of one function becomes the input of other, the function is a composite function.
In mathematics, the composition of a function is a step-wise application. For example, the function f: A→ B & g: B→ C can be composed to form a function which maps x in A to g(f(x)) in C. All sets are non-empty sets. A composite function is denoted by (g o f) (x) = g (f(x)). The notation g o f is read as “g of f”.
Consider the functions f: A→B and g: B→C. f = {1, 2, 3, 4, 5}→ {1, 4, 9, 16, 25} and g = {1, 4, 9, 16, 25} → {2, 8, 18, 32, 50}. A = {1, 2, 3, 4, 5}, B = {16, 4, 25, 1, 9}, C = {32, 18, 8, 50, 2}.Here, g o f = {(1, 2), (2, 8), (3, 18), (4, 32), (5, 50)}.
The composition of functions is associative in nature i.e., g o f = f o g. It is necessary that the functions are one-one and onto for a composition of functions.
Invertible Function
A function is invertible if on reversing the order of mapping we get the input as the new output. In other words, if a function, f whose domain is in set A and image in set B is invertible if f-1 has its domain in B and image in A.
f(x) = y ⇔ f-1 (y) = x.
Not all functions have an inverse. For a function to have an inverse, each element b∈B must not have more than one a ∈ A. The function must be an Injective function. Also, every element of B must be mapped with that of A. The function must be a Surjective function. It is necessary that the function is one-one and onto to be invertible, and vice-versa.
It is interesting to know the composition of a function and its inverse returns the element of the domain.
f-1 o f = f -1 (f(x)) = x
Algebra of Real Functions
Real-valued Mathematical Functions
In mathematics, a real-valued function is a function whose values are real numbers. It is a function that maps a real number to each member of its domain. Also, we can say that a real-valued function is a function whose outputs are real numbers i.e., f: R→R (R stands for Real).
Algebra of Real Functions
In this section, we will get to know about addition, subtraction, multiplication, and division of real mathematical functions with another.
Addition of Two Real Functions
Let f and g be two real valued functions such that f: X→R and g: X→R where X ⊂ R. The addition of these two functions (f + g): X→R is defined by:
(f + g) (x) = f(x) + g(x), for all x ∈ X.
Subtraction of One Real Function from the Other
Let f: X→R and g: X→R be two real functions where X ⊂ R. The subtraction of these two functions (f – g): X→R is defined by:
(f – g) (x) = f(x) – g(x), for all x ∈ X.
Multiplication by a Scalar
Let f: X→R be a real-valued function and γ be any scalar (real number). Then the product of a real function by a scalar γf: X→R is given by:
(γf) (x) = γ f(x), for all x ∈ X.
Multiplication of Two Real Functions
The product of two real functions say, f and g such that f: X→R and g: X→R, is given by
(fg) (x) = f(x) g(x), for all x ∈ X.
Division of Two Real Functions
Let f and g be two real-valued functions such that f: X→R and g: X→R where X ⊂ R. The quotient of these two functions (f ⁄ g): X→R is defined by:
(f / g) (x) = f(x) / g(x), for all x ∈ X.
Note: It is also called pointwise multiplication.
