Standard Real Functions and their Graphs2019-09-14_18-47-31

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 .

 

2019-09-14_18-49-42.jpgIdentity 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.

2019-09-14_18-51-05.jpgModulus 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|

2019-09-14_18-52-20Greatest 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

2019-09-14_18-53-33.jpgSmallest 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} 

2019-09-14_18-54-45Fractional 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 

2019-09-14_18-55-50.jpgSignum 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<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}

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 .

2019-09-14_19-03-22.jpgReciprocal 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 .

2019-09-14_19-07-27.jpgSquare 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 ) .

2019-09-14_19-08-19.jpgSquare 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 ) .

2019-09-14_19-09-24Cube 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.

2019-09-14_19-09-50Cube 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 .