Functions
4.0 Functions
4.0 Functions
Definition:
Let $A$ and $B$ be two non-empty sets. A relation $f$ from $A$ to $B$, i.e. a subset of $A \times B$, is called a function from $A$ to $B$, if
$$\begin{equation} \begin{aligned} i)\;for\;each\;a \in A\;there\;exists\;b \in B\;such\;that\;(a,b) \in f \\ ii)\;(a,b) \in f\;and\;(a,c) \in f \Rightarrow b = c \\\end{aligned} \end{equation} $$
If $(a,b) \in f$ , then $b$ is called the image of $a$ under $f$ and $a$ is called the pre-image of $b$ under the function $f$.
Function as a Correspondence
Definition:
Let $A$ and $B$ be two non-empty sets. Then a function from set $A$ to set $B$ is a rule or correspondence or method which associates elements of set $A$ to elements of set $B$ such that,
$$\begin{equation} \begin{aligned} i)\;all\;elements\;of\;set\;A\;are\;associated\;in\;set\;B \\ ii)\;an\;element\,of\;set\,A\;is\;associated\;to\;a\;unique\;element\;in\;set\;B \\\end{aligned} \end{equation} $$
In other words, a function $f$ from a set $A$ to set $B$ associates each and every element of $A$ to a unique element of set $B$.
Graphical representation of a function:
If any vertical line cuts the graph at more than $1$ point, then the graph does not represent a function as shown in figure.
Description of a function
Let $f:A \to B$ 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.
Basic terms:
1. Domain
Let $f:A \to B$. Then, the set $A$ is known as the domain of $f$.
2. Co-domain
Let $f:A \to B$. Then, the set $B$ is known as the co-domain of $f$.
3. Range
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 $f(A)$.
4. Equal Function
Two functions $f$ and $g$ are said to be equal iff
$$\begin{equation} \begin{aligned} i)\;domain\;off\; = \;domain\;of\;g \\ ii)\,co - domain\;of\;f = co - domain\;of\;g \\ and\;iii)\;f(x) = g(x)\;for\;every\;x\;belonging\;to\;their\;common\;domain. \\\end{aligned} \end{equation} $$
When two functions are equal, they are simply written as $f=g$.
5. Inverse of a function
If $f(x) = y$ then, ${f^{ - 1}}(y) = x$, given that, $f:A \to B,\;y \in B$ and $x \in A$
${f^{ - 1}}(y)$ is the set of pre-images of $y$.
6. Real Valued Functions and real Functions
A function $f:A \to B$ is called a real valued function, if $B$ is a subset of $R$, where $R$ is a set of all real numbers.
When $A \subset R$ and $B \subset R$, then $f$ is called the real function.
- Domain of the real function is the set of all those real values for which the function assumes real values only.
- Range of a real function: The range of a real function is found using the following steps:
1. Consider $f(x) = y$
2. Solve the function and represent $x$ in terms of $y$.
3. Now, since $x$ can be represented in terms of $y$, consider another function $\phi (y) = x$
4. Find all values of $y$ for which values of $x$ obtained from $\phi (y) = x$ is real. Also, the values of $x$ should be within the domain.
5. The set of all values of $y$, obtained, is the range of the function $f$.
Example 13. Find the range of $f(x) = \frac{{{x^2} + x + 1}}{{{x^2} + x - 1}}$.
Solution: We can write the function as $$y = \frac{{{x^2} + x + 1}}{{{x^2} + x - 1}}$$ Now, writing $x$ in terms of $y$ as $$\begin{equation} \begin{aligned} y\left( {{x^2} + x - 1} \right) = {x^2} + x + 1 \\ y{x^2} + yx - y = {x^2} + x + 1 \\ {x^2}\left( {y - 1} \right) + x\left( {y - 1} \right) - y - 1 = 0 \\\end{aligned} \end{equation} $$
From the equation, we can say that if $y=1$, then the above equation becomes $-2=0$ which is never true. Therefore, we get a quadratic equation in $x$ which has real roots. if (using ${b^2} - 4ac \geqslant 0$)),
$$\begin{equation} \begin{aligned} {(y - 1)^2} - 4(y - 1)( - y - 1) \geqslant 0 \\ \left( {{y^2} + 1 - 2y} \right) + 4\left( {{y^2} - 1} \right) \geqslant 0 \\ 5{y^2} - 2y - 3 \geqslant 0 \\ y \leqslant - 3/5\;or\;y \geqslant 1\;but\;y \ne 1 \\\end{aligned} \end{equation} $$
Therefore, range of the function $f(x)$ is $$\left( { - \infty , - \frac{3}{5}} \right] \cup \left( {1,\infty } \right)$$
