Function as a Special Kind of Relation

Definition: Let $A$ and $B$  be two non-empty sets. A relation $f$ from $A$ to $B$, i.e., a sub-set of $A \times B$, is called a function  from $A$ to $B$, if
(i) for each $a \in A$ there exists $b \in B$ such that $(a,b) \in f$
(ii) $(a,b) \in f$ and $(a,c) \in f +b=c$.

Therefore a non-void subset $f$ of $A \times B$ is a function from $A$ to $B$ if each element of $A$ appears in some ordered pair in f $and-no two ordered pairs in $f$ have the same first element. If $(a,b) \in f$ then $b$ is called the image of $a$ under $f$. Function as a Correspondence Definition: Let $A$ and $B$ be two non-empty sets. Then a function $f$ from set $A$ to set $B$ is a rule or method or correspondence which associates elements of set $A$ to elements of set $B$ such that: i) all elements of set $A$ are associated to elements in set $B$. ii) an element of set $A$ is associated to a unique element in set $B$. In other words, a function $'f'$ from a set $A$ to a set $B$ associates each element of $A$ to a unique element of set $B$. Let $f :A \rightarrow B$ be a function such that the set $A$ consists of a finite number of elements.Then, $f(x)$ be described by listing, the values which it attains at different points of its domain. Domain, Co-domain and Range of a Function Let $f :A \rightarrow B$. Then the set $A$ is known as the domain of $f$and the set $B$ is known as the co-domain of $f$. The set of all $f$ images of elements of $A$ is known as the range of $f$ or image set of $A$ under $f$ and is denoted by $f (A)$. Thus, $f (A) = \{ f(x) : x \in A \} = \ Range \ of \ f$ Also $f(A) \subseteq B$ Example: Let $A = \{ -2, -1, 0, 1, 2 \}$ and $B = \{ 0, 1,2, 3,4,5,6 \}$. Consider a rule $f (x) =x^2$. Under this rule, we obtain $f(-2)=(-2)^2 =4, \ \ f(-1)=(-1)^2 =1, \ \ f(0)=0^2 =0, \ \ f (1)=1^2 =1$ and $f (2) =2^2 = 4$. We observe that each element of $A$ is associated to a unique element of $B$. So, $f : A \rightarrow B$ given by $f(x) = x^2$ is a function. Clearly, domain $(f)=A={-2,-1,0,1,2}$ and range $(f)={0,1,4}$. Equal Functions : Two functions $f$ and $g$ are said to be equal if and only if (i) domain of $f =$ domain of $g$ (ii) co-domain of $f =$ co-domain of $g$, and (iii) $f (x) = g(x)$ for every $x$ belonging to their common domain. If two functions $f$ and $g$ are equal, then we write $f = g$. Example: Let $A = \{ 1, 2 \} , B = \{ 3, 6 \}$ and $f : A \rightarrow B$ given by $f (x) = x^2 + 2$ and $g : A \rightarrow B$ given by $g(x) = 3x$. Then, we observe that $f$ and $g$ have the same domain and co-domain. Also we have, $f (1)= 3=g(1)$ and $f (2)=6=g(2)$. Hence, $f=g$. Real Functions: functions having domain and co-domain both as subsets of the set$latex R of all real number. Such functions are called real functions or real valued functions.

Definition: A function $f : A \rightarrow B$ is called a real valued function, if $B$ is a subset of $R$ ( set of all real numbers).

If both $A$ and $B$ both are subsets of $R$, then f is called a real function.

Domain of a Real Function: the domain of the real function $f(x)$ is the set of all those real numbers for which the expression for $f (x)$ or the formula $f(x)$ assumes real values only.

Range of a Real Function: The range of a real function of a real variable is the set of all real values taken by $f (x)$ at points in its domain.

Standard Real Functions and their Graphs (to be linked)

Operations on Real Functions

Addition: If $f : D_1 \rightarrow R$ and $g : D_2 \rightarrow R$ be two real functions. Then, their sum $f + g$ is defined as that function from $D_1 \cap D_2$ to $R$ which associates each $x \in D_1 \cap D_2$ to the number $f (x) + g (x)$.

$(f+g)(x) = f(x) + g(x)$  for all $x \in D_1 \cap D_2$

Product: If $f : D_1 \rightarrow R$ and $g : D_2 \rightarrow R$ be two real functions. Then, their product $fg$ is a function from $D_1 \cap D_2$ to $R$ and is defined as

$(fg) (x) = f(x) g(x)$  for all $x \in D_1 \cap D_2$

Subtraction: If $f : D_1 \rightarrow R$ and $g : D_2 \rightarrow R$ be two real functions. Then, their difference $f - g$ is defined as

$(f-g)(x) = f(x) - g(x)$  for all $x \in D_1 \cap D_2$

Quotient: If $f : D_1 \rightarrow R$ and $g : D_2 \rightarrow R$ be two real functions. Then, their quotient $\frac{f}{g}$  is a function from $D_1 \cap D_2 - \{ x:g(x)=0\}$ to $R$ and is defined as

$\Big($ $\frac{f}{g}$ $\Big) (x) =$ $\frac{f(x)}{g(x)}$ for all $x \in D_1 \cap D_2 - \{ x:g(x)=0\}$

Multiplication of a Function by a Scalar: Let $f : D \rightarrow R$ be a real function and $\alpha$ be a scalar (real number). Then the product $\alpha f$ is a function from $D$ to $R$ and is defined as

$(\alpha f) (x) = \alpha f (x)$ for all $x \in D$

Reciprocal of a Function: If $f \colon D \rightarrow R$ is a real function , then its reciprocal function $\frac{1}{f}$  is a function from $D - \{ x: f(x) = 0 \}$ to $R$ and is defined as

$\Big($ $\frac{1}{f}$ $\Big) (x) =$ $\frac{1}{f(x)}$