Question

Which of the following is true for $y=|1-\frac{1}{|x|-2}|?$

• A. It has $2$ asymptotes.
  
• B. It has $3$ asymptotes.
  
• C. It is not differentiable at $5$ points.
  
• D. It is not differentiable at $2$ points.

Answer 1

Answer: (B), (D)

The correct options are
B It has $3$ asymptotes.

D It is not differentiable at $2$ points.

Follow the below steps to draw the graph of function $y=|1-\frac{1}{|x|-2}|$

Step1: First draw the graph of function $y=\frac{1}{x-2}$

Step 2: Now draw the graph for function $y=\frac{1}{|x|-2}$

Step 3: Draw the graph of $y=-\frac{1}{|x|-2}$

Step 4: Draw the graph of $y=1-\frac{1}{|x|-2}$

Step 5: Now draw the graph of $y=|1-\frac{1}{|x|-2}|$

From graph of step 5, it is clear that it has $3$ asymptotes and not differentiable at $2$ points.