Standard Real Functions and their Graphs

Constant Function: If $k$ is a fixed real number, then a function $f (x)$ given by $f (x) = k$ for all $x \in R$ is called a constant function. Sometimes we also call it the constant function $k$.

Identity Function: The function that associates each, real number to itself is called the identity function and is usually denoted by $I$. Thus, the function $I \colon R \rightarrow R$ defined by $I (x) = x$ for all $x \in R$ is called the identity function.

Clearly, the domain and range of the identity function are both equal to $R$. The graph of the identity function is a straight line passing through the origin and inclined at an angle of $45^o$ with X-axis.

Modulus Function: The function $f (x)$ defined by $f (x) = | x | = \Big\{ \begin{array}{cc} x, \hspace*{0.5cm} when \ x \geq 0 \\ -x, \hspace*{0.4cm} when \ x < 0 \end{array}$ is called modulus function.

It is also called the absolute value function. We observe that the domain of the modulus function is the set $R$ of all real numbers and the range is the set of all non-negative real numbers i.e. $R^+ = \{ x \in R : x \geq 0 \}$

Properties of Modulus Function:

i) For any real number $x, \sqrt{x^2} = |x|$

ii) If $a,b$ are positive real numbers, then

$x^2 \leq a^2 \Leftrightarrow |x| \leq a \Leftrightarrow -a \leq x \leq a$

$x^2 \geq a^2 \Leftrightarrow |x| \geq a \Leftrightarrow x \leq -a \ or \ x \geq a$

$x^2 < a^2 \Leftrightarrow |x| < a \Leftrightarrow -a < x < a$

$x^2 > a^2 \Leftrightarrow |x| > a \Leftrightarrow x < -a \ or \ x > a$

$a^2 \leq x^2 \leq b^2 \Leftrightarrow a \leq |x| \leq b \Leftrightarrow x \in [ -b, -a ] \cup [a,b]$

$a^2 < x^2 < b^2 \Leftrightarrow a < |x| < b \Leftrightarrow x \in ( -b, -a ) \cup (a,b)$

iii) For real numbers x and y, we have

$| x+y | = |x| + |y| \Leftrightarrow ( x \geq 0 \ and \ y \geq 0) \ or \ ( x < 0 \ and \ y < 0)$

$| x-y | = |x| - |y| \Leftrightarrow ( x \geq 0, y \geq 0 \ and \ |x| \geq |y| ) \ or \ ( x \leq 0, y \leq 0 \ and \ |x| \geq |y|)$

$|x \pm y | \geq |x| + |y|$

$|x \pm y | \geq |x| - |y|$

Greatest Integer Function: For any real number & we use the symbol $[x]$ or, $\lfloor x \rfloor$ to denote the greatest integer less than or equal to $x$.

The function $f:R \rightarrow R$ defined by $f (x) =[x]$ for all $x \in R$ is called the greatest integer function or the floor function.

It is also called a step function. Clearly, domain of the greatest integer function is the set $R$ of all real numbers and the range is the set $Z$ of all integers as it attains only integer values.

Properties of Greatest Integer Function: If $n$ is an integer and $x$ is a real number between $n$ and $n + 1$, then:

i) $[-n] = [n]$

ii) $[x+k] = [x] +k$ for any integer $k$

iii) $[-x] = -[x]-1$

iv) $[x] +[-x] = \Big\{ \begin{array}{ll} -1, \hspace*{0.7cm} \ if \ x \notin Z \\ 0, \hspace*{1.0cm} \ if x \in Z \end{array}$

v) $[x] - [-x] = \Big\{ \begin{array}{ll} 2[x]+1, \hspace*{0.3cm} \ if \ x \notin Z \\ 2[x], \hspace*{0.9cm} \ if \ x \in Z \end{array}$

vi) $[x] \geq k \Rightarrow x \geq k, \ where \ k \in Z$

vii) $[x] \geq k \Rightarrow x < k < 1 , \ where \ k \in Z$

viii) $[x] > k \Rightarrow x > k+1, \ where \ k \in Z$

ix)  $[x] < k \Rightarrow x < k , \ where \ k \in Z$

x) $[x+y] = [x] + [y+x-[x]] \ for \ all \ x, y \in R$

xi) $[x] + [x +$ $\frac{1}{n}$ $] + [x +$ $\frac{2}{n}$ $] + \ldots + [x +$ $\frac{n-1}{n}$ $] = [nx], n \in N$

Smallest Integer Function: The function $f:R \rightarrow R$ defined by $f (x) = \lceil x \rceil$ for all $x \in R$ is called the smallest integer function or the ceiling function.

It is also a step function. We observe that the domain of the smallest integer function is the set $R$ of all real numbers and its range is the set $Z$ of all integers.

Properties of Smallest Integer Function: Following are some properties of smallest
integer function:

i) $\lceil -n \rceil = - \lceil n \rceil , \ where \ n \in Z$

ii) $\lceil -x \rceil = - \lceil x \rceil + 1, \ where \ x \in R-Z$

iii) $\lceil x + n \rceil = \lceil x \rceil + n, \ where \ x \in R-Z \ and \ n \in Z$

iv) $\lceil x \rceil + \lceil -x \rceil = \Big\{ \begin{array}{ll} 1, \ if \ x \notin Z \\ 0, \ if \ x \in Z \end{array}$

v) $\lceil x \rceil + \lceil -x \rceil = \Big\{ \begin{array}{ll} 2 \lceil x \rceil - 1, \ if \ x \notin Z \\ 2 \lceil x \rceil , \ if \ x \in Z \end{array}$

Fractional Part Function: For any real number $x$ we use the symbol $\{ x \}$ to denote the fractional part or decimal part of $x$.

The function $f :R \rightarrow R$ defined by $f (x) = \{ x \}$ for all $x \in R$ is called the fractional part function. The domain of the fractional part function is the set $R$ of all real numbers and the range of the set $[0, 1)$.

Therefore from the definition, $f(x) = \{ x \} = x - [x]$ for all $x \in R$

Signum Function: The function $f$ defined by:

$f(x) = \Bigg\{ \begin{array}{ll} \frac{|x|}{x} \hspace*{0.5cm} x \neq 0 \\ 0 \hspace*{0.6cm} x = 0 \end{array}$    or    $f(x) = \Bigg\{ \begin{array}{lll} 1 \hspace*{0.7cm} x > 0 \\ 0, \hspace*{0.7cm} x = 0 \\ -1 \hspace*{0.5cm} x < 0 \end{array}$

The domain of the signum function is the set $R$ of all real numbers and the range is the set of $\{ -1, 0, 1 \}$

Exponential Function: If $a$ is a positive real number other than unity, then a function that associates each $x \in R$ to $a^x$ is called the exponential function.

Or a function $f:R \rightarrow R$ defined by $f (x)=a^x$, where $a>0$ and $a\neq 1$ is called the exponential function.

The domain of an exponential function is $R$ the set of all real numbers and and the range is the set $(0, \infty)$ as it attains only positive values.

Case 1: When $a>1$, the values of $y = f (x) = a^x$ increase as the values of $x$ increase.

$f(x) = a^x \Bigg\{ \begin{array}{lll} < 1 \hspace*{0.5cm} \ for \ x < 0 \\ =1 \hspace*{0.5cm} \ for \ x = 0 \\ > 1 \hspace*{0.5cm} \ for \ x > 0 \end{array}$

Case 2: When $0 < a < 1$, In this case, the values of $y=f(x)=a^x$ decrease with the increase in $x$ and $y>0$ for all $x \in R$.

$f(x) = a^x \Bigg\{ \begin{array}{lll} > 1 \hspace*{0.5cm} \ for \ x < 0 \\ =1 \hspace*{0.5cm} \ for \ x = 0 \\ < 1 \hspace*{0.5cm} \ for \ x > 0 \end{array}$

Logarithmic Function: If $a > 0$ and $a \neq 1$ , then the function defined by $f (x) = \log_a x, x > 0$ is called logarithmic function.

Note that logarithmic function and the exponential function are inverse functions i.e. $\log_a x = y \Leftrightarrow x = a^y$

The domain of the logarithmic function is the set of all non-negative real numbers i.e. $(0, \infty)$ and the range is the set R of all real numbers.

Case 1: When $a>1$

$f(x) = \log_a x \Bigg\{ \begin{array}{lll} < 0 \hspace*{0.5cm} \ for \ x < 0 < 1 \\ =0 \hspace*{0.5cm} \ for \ x = 1 \\ > 1 \hspace*{0.5cm} \ for \ x > 1 \end{array}$

The values of $y$ increase with the increase in $x$.

Case 2:  When $0 < a < 1$

$f(x) = \log_a x \Bigg\{ \begin{array}{lll} > 0 \hspace*{0.5cm} \ for \ x < 0 < 1 \\ =0 \hspace*{0.5cm} \ for \ x = 1 \\ < 0 \hspace*{0.5cm} \ for \ x > 1 \end{array}$

The values of $y$ decrease with the increase in $x$.

Properties of Logarithmic Functions:

i) $\log_a 1 = 0$  where $a > 0 , a \neq 1$

ii) $\log_a a = 0$  where $a > 0 , a \neq 1$

iii) $\log_a (xy) = \log_a |x| + \log_a |y|$, where $a > 0, a \neq 1$ and $xy > 0$

iv) $\log_a \Big( \frac{x}{y} \Big) = \log_a |x| - \log_a |y|$ , where $a > 0, a \neq 1$ and $\frac{x}{y}$ $> 0$

v) $\log_a (x^n) = n \log_a |x|$ , where $a > 0, a \neq 1$  and $x^n > 0$

vi) $\log_{a^n} x^m = \frac{m}{n} \log_a x$ , where $a > 0, a \neq 1$ and $x > 0$

vii) $x^{\log_a y} = y^{\log_a x}, where x > 0, y > 0 a > 0 , a \neq 1$

viii) If $a >1$, then the values of $f (x) = \log_a x$ increase with the increase in $x$.

$x

Also $f(x) = \log_a x \Bigg\{ \begin{array}{lll} < 0 \hspace*{0.5cm} \ for \ x < 0 < 1 \\ =0 \hspace*{0.5cm} \ for \ x = 1 \\ > 0 \hspace*{0.5cm} \ for \ x > 1 \end{array}$

ix) If $0 < a < 1$, then the values of $f (x) = \log_a x$ decrease with the increase in $x$. i.e $x < y \Leftrightarrow \log_a x > \log_a y$

Also $f(x) = \log_a x \Bigg\{ \begin{array}{lll} > 0 \hspace*{0.5cm} \ for \ x < 0 < 1 \\ =0 \hspace*{0.5cm} \ for \ x = 1 \\ < 0 \hspace*{0.5cm} \ for \ x > 1 \end{array}$

x) $\log_a x =$ $\frac{1}{\log_x a}$ for $a > 0, a \neq 1$ and $x > 0 , x \neq 1$

Note: Functions $f (x) = \log_a x$ and $g (x) = a^x$ are inverse of each other. So, their graphs are mirror images of each other in the line mirror $y = x$.

Reciprocal Function: The function that associates a real number $x$ to its reciprocal $\frac{1}{x}$ is called the reciprocal function. Since $\frac{1}{x}$ is not defined for $x=0$ we define the reciprocal function as follows:

The function $f \colon R - \{ 0 \} \rightarrow R$  defined by $f(x) =$ $\frac{1}{x}$ is called a reciprocal function.

The domain of the reciprocal function is $R - \{ 0 \}$ and its range is also $R - \{ 0 \}$.

The sign of $\frac{1}{x}$ is the same as that of $x$ and $\frac{1}{x}$ decreases with the increase in $x$.

Square Root Function: The function that associates a real number $x$ to $+ \sqrt{x}$ is called the square root function. Since $\sqrt{x}$ is real for $x > 0$. So, we defined the square root function as follows:

The function $f :R \rightarrow R$ defined by $f (x) = + \sqrt{x}$ is called the square root function.

The domain of the square root function is $R^+$ i.e. $[0, \infty )$ and its range is also $[0, \infty )$.

Square Function: The function that associates a real number $x$ to its square i.e. $x^2$ is called the square function. Since $x^2$ is defined for all $x \in R$. So, we define the square function as follows:

The function $f :R \rightarrow R$ defined by $f (x)=x^2$ is called the square function. Clearly, domain of the square function is $R$ and its range is the set of all non-negative real numbers i.e. $[0, \infty )$.

Cube Function: The function that associate a real number $x$  to its cube is called the cube function. If $x^3$ is meaningful for all $x \in R$  we define the cube function as follows:

The function $f :R \rightarrow R$ defined by $f (x)=x^3$ is called the cube function.

The sign of $x$ and $x^3$ will always be the same. When $x$ increases, $x^3$ increase too. The graph $f(x) = x^3$ is symmetrical  in opposite quadrant.

Cube Root Function: The function that associate a real number $x$  to its cube root $x^{\frac{1}{3}}$ is called the cube root function

The function $f :R \rightarrow R$ defined by $f (x)=x^{\frac{1}{3}}$ is called the cube function.

Domain and range of the cube root function are both equal to $R$.