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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $f$ and $g$ be two functions defined by

$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\|x-1|, & x \geq 0\end{array} \text { and } g(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\1, & x \geq 0\end{array}\right. \text {. }\right.$

Then (gof) (x) is

  • A
    Differentiable everywhere
  • ✓
    Continuous everywhere but not differentiable exactly at one point
  • C
    Not continuous at $x =-1$
  • D
    Continuous everywhere but not differentiable at $x=1$
✓

Answer

Correct option: B.
Continuous everywhere but not differentiable exactly at one point
b
$f(x)=\left\{\begin{array}{c}x+1, \quad x < 0 \\ 1-x, \quad 0 \leq x<1 \\ x-1,1 \leq x\end{array}\right.$

$g(x)=\left\{\begin{array}{c} x +1, x < 0 \\ 1, x \geq 0\end{array}\right.$

$g(f(x))=\left\{\begin{array}{c} x +2, x < -1 \\ 1, x \geq-1\end{array}\right.$

$g(f(x))=\left\{\begin{array}{c} x +2, x < -1 \\1, x \geq-1\end{array}\right.$

$\therefore g ( f ( x ))$ is continuous everywhere

$g ( f ( x ))$ is not differentiable at $x =-1$

Differentiable everywhere else

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