Functions
A function is a special case of a relation. To be specific, let X, Y be two non-empty sets and R be a relation from X to Y, then
R may not relate an element of X to an element of Y or
R may relate an element of X to more than one element of Y
But a function relates each element of X to a unique element of Y.
Definition: If X,Y are two non-empty sets then a subset of X⨯Y is called a function(or mapping or map) from X to Y iff for each x∈X, there exists a unique y∈Y such that (x,y)∈f.
A function from X to Y is denoted by f:X→Y
The unique element y is called the image of the element x under the function f:X→Y. It is denoted by f(x) i.e. y=f(x). The element y is also called the value of the function f at x.
If f(x)=y, then x is a pre-image of y.
Thus, a subset f of X⨯Y is called a function from X to Y iff
(i) All elements of set X are associated with the elements of set Y
(ii) An element of set X is associated with one and only one element of set Y.
f:X→Y such that {(x,f(x)): x∈X and f(x)∈Y}
Domain, Range, Co-domain
Let f be a function from X to Y, then
Set X is called the domain of the function f.
Set Y is called the Co-domain of the function f.
Set consisting of all the images of the elements of X is called the range of function f.
range of f={f(x): for all x∈X}
range of f is a subset of Y (co-domain) which may or may not be equal to Y.
Features of a function
let f be a function from A to B, then
(i) to every x∈A, there exists a unique element y∈B such that y=f(x)
(ii) no element of A can have more than one image in B
(iii) there may be elements of B which are not associated with any element of A
(iv) distinct elements of A may have same image in B
(v) function f is determined when f(x) is known for all x∈A
Real Functions
A function which has either R or one of its subset as its range is called a real valued function. Further, if its domain is also either R or a subset of R, then its called a real function.
Domain and Range of a real function
Let f be a real function, then
The set of all possible real numbers x for which f(x) is a real number is called the domain of the function f and is denoted by $\displaystyle \small D_{f}$.
$\displaystyle \small D_{f}$={x: x∈R, f(x)∈R}
The set of images f(x) for all x∈$\displaystyle \small D_{f}$ is called the range of f and is denoted by $\displaystyle \small R_{f}$.
$\displaystyle \small R_{f}$={f(x): for all x∈$\displaystyle \small D_{f}$}
A function does not exist if its domain is empty set.
Equal Functions
Two functions f and g are called equal functions, written as f=g, if and only if
(i) domain of f = domain of g and
(ii) f(x) = g(x) for all x in domain of f (or g)
Some functions and their graphs
1. Constant Function
A function f:A→B, (A,B⊂R), is said to be a constant function if there exists a real number k such that f(x)=k for all x∈A.
(i) Domain is R
(ii) Range is {k}
(iii) If k>0, then graph will be a parallel line above x-axis
(iv) If k=0, then graph is x-axis itself and is called zero function
(v) If k<0, then graph will be a parallel line below x-axis
2. Identity Function
The function f:R→R defined by f(x)=x for each x∈R, is called the identity function.
(i) Domain is R
(ii) Range is R
(iii) The graph is a straight line
(iv) It passes through the origin
(v) Its slope is 1
3. Polynomial Function
A function f:R→R, defined by $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n}$ where n∈N; $\displaystyle \small a_{0}$,$\displaystyle \small a_{1}$,$\displaystyle \small a_{2}$...$\displaystyle \small a_{n}$∈R, is called a polynomial function.
Example: f(x)=$\displaystyle \small x^{3}$-$\displaystyle \small x^{2}$+2
4. Rational Function
A function of the form $f(x)=\frac{p(x)}{q(x)}$ where p(x) and q(x) are polynomials and q(x)≠0 is called a rational function.
Example: $f(x)=\left ( \frac{x^{2}+1}{x^{3}-2x+5} \right )$
Its domain is R-{x:$\displaystyle \small x^{3}$-2x+5≠0}
5. The Modulus Function
The function f:R→R defined by
$f(x)=|x|=\left\{\begin{matrix} x & if & x\geq 0\\ -x & if & x< 0 \end{matrix}\right.$
is called the modulus function. It is also called absolute value function.
(i) Domain is R
(ii) Range is [0,∞)
(iii) the graph is symmetric with respect to the y-axis
(iv) it is above the x-axis, except at one point, x=0
6. Signum Function
The function f:R→R defined by
$f(x)=\left\{\begin{matrix} \frac{|x|}{x} & when & x\neq 0\\ 0 & when & x= 0 \end{matrix}\right.$
Thus $f(x)=\left\{\begin{matrix} 1 & when &x> 0 \\ 0 & when &x= 0 \\ -1 & when &x< 0 \end{matrix}\right.$
(i) Domain is R
(ii) Range is {-1,0,1}
7. Greatest Integer Function
The function f:R→R defined by f(x)=[x] is called the greatest integer function, where [x]=integral part of x or greatest integer less than or equal to x.
i.e. f(x)=n, where n≤x<n+1, n∈I (the set of integers)
It is also called floor function or step function or integral function
[x]= -1 for -1≤x<0
[x]= 0 for 0≤x<1
[x]= 1 for 1≤x<2
[x]= 2 for 2≤x<3 and so on.
(i) Domain is R
(ii) Range is I
Algebra of Real Functions
The algebraic operations of addition, subtraction, multiplication and division etc. can be performed on two real valued functions.
Let f:X→R and g:X→R be any two real functions where X⊂R, then
(i) Addition
The sum of f and g denoted by f+g, is the function defined by
(f+g)(x)= f(x)+g(x), for all x∈X
(ii) Subtraction
The difference of f and g denoted by f-g, is the function defined by
(f-g)(x)= f(x)-g(x), for all x∈X
(iii) Multiplication by a scalar
If c is any real number, then scalar multiple of f by c is defined as
(cf)(x) = c f(x), for all x∈X
(iv) Multiplication
The product of f and g denoted by fg, is the function defined by
(fg)(x)= f(x)g(x), for all x∈X
(v) Quotient
The quotient of f by g denoted by $\frac{f}{g}$ , is the function defined by
$\left ( \frac{f}{g} \right )\left ( x \right )=\frac{f(x)}{g(x)}$ , for all x∈X
Example: f(x)=$\displaystyle \small x^{2}$ and g(x)=2x+1
(f+g)(x)= f(x)+g(x) = $\displaystyle \small x^{2}$+2x+1
(f-g)(x)= f(x)-g(x) = $\displaystyle \small x^{2}$-2x-1
(4g)(x) = 4 g(x) = 8x+8
(fg)(x)= f(x)g(x) = $\displaystyle \small x^{2}$(2x+1) = 2$\displaystyle \small x^{3}$+$\displaystyle \small x^{2}$
$\left ( \frac{f}{g} \right )\left ( x \right )=\frac{f(x)}{g(x)}$ = $\frac{x^{2}}{2x+1}$
A function is a special case of a relation. To be specific, let X, Y be two non-empty sets and R be a relation from X to Y, then
R may not relate an element of X to an element of Y or
R may relate an element of X to more than one element of Y
But a function relates each element of X to a unique element of Y.
Definition: If X,Y are two non-empty sets then a subset of X⨯Y is called a function(or mapping or map) from X to Y iff for each x∈X, there exists a unique y∈Y such that (x,y)∈f.
A function from X to Y is denoted by f:X→Y
The unique element y is called the image of the element x under the function f:X→Y. It is denoted by f(x) i.e. y=f(x). The element y is also called the value of the function f at x.
If f(x)=y, then x is a pre-image of y.
Thus, a subset f of X⨯Y is called a function from X to Y iff
(i) All elements of set X are associated with the elements of set Y
(ii) An element of set X is associated with one and only one element of set Y.
f:X→Y such that {(x,f(x)): x∈X and f(x)∈Y}
Domain, Range, Co-domain
Let f be a function from X to Y, then
Set X is called the domain of the function f.
Set Y is called the Co-domain of the function f.
Set consisting of all the images of the elements of X is called the range of function f.
range of f={f(x): for all x∈X}
range of f is a subset of Y (co-domain) which may or may not be equal to Y.
Features of a functionlet f be a function from A to B, then
(i) to every x∈A, there exists a unique element y∈B such that y=f(x)
(ii) no element of A can have more than one image in B
(iii) there may be elements of B which are not associated with any element of A
(iv) distinct elements of A may have same image in B
(v) function f is determined when f(x) is known for all x∈A
Real Functions
A function which has either R or one of its subset as its range is called a real valued function. Further, if its domain is also either R or a subset of R, then its called a real function.
Domain and Range of a real function
Let f be a real function, then
The set of all possible real numbers x for which f(x) is a real number is called the domain of the function f and is denoted by $\displaystyle \small D_{f}$.
$\displaystyle \small D_{f}$={x: x∈R, f(x)∈R}
The set of images f(x) for all x∈$\displaystyle \small D_{f}$ is called the range of f and is denoted by $\displaystyle \small R_{f}$.
$\displaystyle \small R_{f}$={f(x): for all x∈$\displaystyle \small D_{f}$}
A function does not exist if its domain is empty set.
Equal Functions
Two functions f and g are called equal functions, written as f=g, if and only if
(i) domain of f = domain of g and
(ii) f(x) = g(x) for all x in domain of f (or g)
Some functions and their graphs
1. Constant Function
A function f:A→B, (A,B⊂R), is said to be a constant function if there exists a real number k such that f(x)=k for all x∈A.
(i) Domain is R
(ii) Range is {k}
(iii) If k>0, then graph will be a parallel line above x-axis
(iv) If k=0, then graph is x-axis itself and is called zero function
(v) If k<0, then graph will be a parallel line below x-axis
2. Identity Function
The function f:R→R defined by f(x)=x for each x∈R, is called the identity function.
(i) Domain is R
(ii) Range is R
(iii) The graph is a straight line
(iv) It passes through the origin
(v) Its slope is 1
3. Polynomial Function
A function f:R→R, defined by $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n}$ where n∈N; $\displaystyle \small a_{0}$,$\displaystyle \small a_{1}$,$\displaystyle \small a_{2}$...$\displaystyle \small a_{n}$∈R, is called a polynomial function.
Example: f(x)=$\displaystyle \small x^{3}$-$\displaystyle \small x^{2}$+2
4. Rational Function
A function of the form $f(x)=\frac{p(x)}{q(x)}$ where p(x) and q(x) are polynomials and q(x)≠0 is called a rational function.
Example: $f(x)=\left ( \frac{x^{2}+1}{x^{3}-2x+5} \right )$
Its domain is R-{x:$\displaystyle \small x^{3}$-2x+5≠0}
5. The Modulus Function
The function f:R→R defined by
$f(x)=|x|=\left\{\begin{matrix} x & if & x\geq 0\\ -x & if & x< 0 \end{matrix}\right.$
is called the modulus function. It is also called absolute value function.
(i) Domain is R
(ii) Range is [0,∞)
(iii) the graph is symmetric with respect to the y-axis
(iv) it is above the x-axis, except at one point, x=0
6. Signum Function
The function f:R→R defined by
$f(x)=\left\{\begin{matrix} \frac{|x|}{x} & when & x\neq 0\\ 0 & when & x= 0 \end{matrix}\right.$
Thus $f(x)=\left\{\begin{matrix} 1 & when &x> 0 \\ 0 & when &x= 0 \\ -1 & when &x< 0 \end{matrix}\right.$
(i) Domain is R
(ii) Range is {-1,0,1}
7. Greatest Integer Function
The function f:R→R defined by f(x)=[x] is called the greatest integer function, where [x]=integral part of x or greatest integer less than or equal to x.
i.e. f(x)=n, where n≤x<n+1, n∈I (the set of integers)
It is also called floor function or step function or integral function
[x]= -1 for -1≤x<0
[x]= 0 for 0≤x<1
[x]= 1 for 1≤x<2
[x]= 2 for 2≤x<3 and so on.
(i) Domain is R
(ii) Range is I
Algebra of Real Functions
The algebraic operations of addition, subtraction, multiplication and division etc. can be performed on two real valued functions.
Let f:X→R and g:X→R be any two real functions where X⊂R, then
(i) Addition
The sum of f and g denoted by f+g, is the function defined by
(f+g)(x)= f(x)+g(x), for all x∈X
(ii) Subtraction
The difference of f and g denoted by f-g, is the function defined by
(f-g)(x)= f(x)-g(x), for all x∈X
(iii) Multiplication by a scalar
If c is any real number, then scalar multiple of f by c is defined as
(cf)(x) = c f(x), for all x∈X
(iv) Multiplication
The product of f and g denoted by fg, is the function defined by
(fg)(x)= f(x)g(x), for all x∈X
(v) Quotient
The quotient of f by g denoted by $\frac{f}{g}$ , is the function defined by
$\left ( \frac{f}{g} \right )\left ( x \right )=\frac{f(x)}{g(x)}$ , for all x∈X
Example: f(x)=$\displaystyle \small x^{2}$ and g(x)=2x+1
(f+g)(x)= f(x)+g(x) = $\displaystyle \small x^{2}$+2x+1
(f-g)(x)= f(x)-g(x) = $\displaystyle \small x^{2}$-2x-1
(4g)(x) = 4 g(x) = 8x+8
(fg)(x)= f(x)g(x) = $\displaystyle \small x^{2}$(2x+1) = 2$\displaystyle \small x^{3}$+$\displaystyle \small x^{2}$
$\left ( \frac{f}{g} \right )\left ( x \right )=\frac{f(x)}{g(x)}$ = $\frac{x^{2}}{2x+1}$
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