Question

 

   

Determine the form of g(x) = f ( f (x)) and hence find the point of discontinuity if g, if any.

Solution

Correct option is

x Ïµ [0, 1]  (1, 2)

           

           

   

   

      

Hence,

          

Now, if (1 + x) Ïµ [1, 2] then   

          f (1 – x) = 1 + (1 + x) = 2 + x             … (i)  

[from original definition of f (x)]  

Similarly, if (1 + x) Ïµ (2, 3) then

          f (1 + x) = 3 – (1+ x) = 2 – x               … (ii)   

If (3 – x) ≤ (0, 1) 

          f (3 – x) = 1 + (3 – x) = 4 – x             … (iii)

Using (i), (ii) and (iii), we get

             

Here, as g(x) changes the inequality sign at x = 1 and x = 2.

Thus, to check continuity at x = 1 and x = 2. 

Now we will check the continuity of g(x) at x = 1, 2

At x = 1 

             

          

As LHL ≠ RHL, g(x) is discontinuous at x = 1.  

At x = 2  

          

           

As. LHL ≠ RHL, g(x) is discontinuous at x = 2.

Thus, g(x) is continuous for all x Ïµ [0, 1] ∪ (1, 2)

SIMILAR QUESTIONS

Q1

Let y = (x) be defined parametrically as y = t2 + t |t|x = 2t – |t|t Ïµ R Then at x = 0, find (x) and discuss continuity.

Q2

 for what value of kf (x) is continuous at x = 0?

Q3

      

Determine a and b such that f (x) is continuous at x = 0.

Q4

Find the points of discontinuity of 

Q5

The left hand derivative of f (x) = [x] sin (πx) at x = kk is an integer, is:

Q6

Which of the following functions is differentiable at x = 0?

Q7

Let f (x) = [n + p sin x], x Ïµ (0, π), n Ïµ Z and p is a prime number, where [.] denotes the greatest integer function. Then find the number of points where f (x) is not Differential.

Q8

, then draw the graph of f (x) in the interval [–2, 2] and discuss the continuity and differentiability in [–2, 2]

Q9

Fill in the blank, statement given below let . The set of points where f (x) is twice differentiable is ……………. .

Q10

The function f (x) = (x2 – 1) |x2 – 3x +2| + cos ( | | ) is not differentiable at