Determine The Form Of g(x) = f ( f (x)) And Hence Find The Point Of Discontinuity If g, If Any.

Why Kaysons ?

Video lectures

Access over 500+ hours of video lectures 24*7, covering complete syllabus for JEE preparation.

Online Support

Practice over 30000+ questions starting from basic level to JEE advance level.

Live Doubt Clearing Session

Ask your doubts live everyday Join our live doubt clearing session conducted by our experts.

National Mock Tests

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.

Organized Learning

Proper planning to complete syllabus is the key to get a decent rank in JEE.

Test Series/Daily assignments

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.

SPEAK TO COUNSELLOR ? CLICK HERE

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