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  • Let  be a function defined by <img alt="f(x)=\frac{x\left [ x \right ]}{1+x^{2}}," src="https://learn.careers360.com/latex

Let f:(1,3)\rightarrow R be a function defined by f(x)=\frac{x\left [ x \right ]}{1+x^{2}}, where \left [ x \right ] denotes the greatest integer \leq x. Then the range of f is :
Option: 1 \left ( \frac{2}{3},\frac{3}{5} \right ]\cup \left ( \frac{3}{4},\frac{4}{5} \right )      
   
Option: 2 \left ( \frac{2}{5},\frac{4}{5} \right ]

Option: 3 \left ( \frac{3}{5},\frac{4}{5} \right )

Option: 4 \left (\frac{2}{5},\frac{1}{2} \right )\cup \left ( \frac{3}{5},\frac{4}{5} \right ]
 

Answers (1)

best_answer

 

 

Piecewise function -


Greatest integer function

The function f: R \small \rightarrow R defined by f(x) = [x], x \small \in R assumes the  value of the greatest integer less than or equal to x. Such a functions called the greatest integer function.

eg;

[1.75] = 1

[2.34] = 2

[-0.9] = -1

[-4.8] = -5


 

From the definition of [x], we

can see that

[x] = –1 for –1 \small \leq x < 0

[x] = 0 for 0 \small \leq x < 1

[x] = 1 for 1 \small \leq x < 2

[x] = 2 for 2 \small \leq x < 3 and

so on.

Properties of greatest integer function:

i) [x] ≤ x < [x] + 1

ii) x - 1 < [x] < x

iii) I ≤ x < I+1 ⇒ [x] = I where I belongs to integer.

iv) [[x]]=[x]v)

v) [x] + [-x] = \left\{\begin{matrix} 0, &if &x & belongs \; to & integer \\ -1, & if &x & doesn't \;belongs \;to & integer \end{matrix}\right.

 

vi) [x] + [-x] = 2x if x belongs to integer

          2[x] + 1 if x doesn’t belongs to integer

-

 

 

Domain of function, Co-domain, Range of function -

All possible values of x for f(x) to be defined is known as a domain. If a function is defined from A to B i.e. f: A?B, then all the elements of set A is called Domain of the function. 

 

If a function is defined from A to B i.e. f: A?B, then all the elements of set B are called Co-domain of the function. 

 

The set of all possible values of  f(x) for every x belongs to the domain is known as Range.

 

For example, let A = {1, 2, 3, 4, 5} and B = {1, 4, 8, 16, 25, 64, 125}. The function f : A -> B is defined by f(x) = x3. So here,

Domain : Set A

Co-Domain : Set B

Range : {1, 8, 27, 64, 125}

 

The range can be equal to or less than codomain but cannot be greater than that.

-

 

 

 

 

\\f(x)=\left\{\begin{array}{ll} {\frac{x}{x^{2}+1} ;} & {x \in(1,2)} \\ {\frac{2 x}{x^{2}+1} ;} & {x \in[2,3)} \end{array}\right.\\\therefore \text{f(x) is a decreasing function }\\\begin{aligned} &\therefore \quad \mathrm{y} \in\left(\frac{2}{5}, \frac{1}{2}\right) \cup\left(\frac{6}{10}, \frac{4}{5}\right]\\ &\Rightarrow \quad y \in\left(\frac{2}{5}, \frac{1}{2}\right) \cup\left(\frac{3}{5}, \frac{4}{5}\right] \end{aligned}

Correct Option (4)

Posted by

vishal kumar

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