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STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 5. continuity and differentiation1 Mark
MCQ
Consider the function $f(x)=\left\{\begin{array}{cc}\frac{x+5}{x-2}, & \text { if } x \neq 2 \\1, & \text { if } x=2\end{array}\right.$ Then, $f(f(x))$ is discontinuous
  • A
    at all real numbers
  • ✓
    at exactly two values of $x$
  • C
    at exactly one value of $x$
  • D
    at exactly three values of $x$
✓

Answer

Correct option: B.
at exactly two values of $x$
b
(b)

We have,

$f(x)=\left\{\begin{array}{cc}\frac{x+5}{x-2}, & x \neq 2 \\1, & x=2\end{array}\right.$

$f(x)$ is discontinuous at $x=2$

$\therefore f(f(x))$ is also discontinuous at $x=2$

Now, $f(f(x))=\frac{f(x)+5}{f(x)-2}=\frac{\frac{x+5}{x-2}+5}{\frac{x+5}{x-2}-2}$

$\Rightarrow f(f(x))=\frac{6 x-5}{9-x}$

Clearly $f(f(x))$ is discontinuous at $x=9$

$\therefore f(f(x))$ is discontinuous at $x=2$ and $9$

Hence, $f(f(x))$ is discontinuous at exactly two values of $x$.

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