GATE (TF) Textile 2018 Question Paper Solution | GATE/2018/TF/22

Question 22 (Textile Technology & Fibre Science)

If $f(x)=\begin{cases} 4-x, & x\leq 2 \\ kx-4, & x> 2 \end{cases}$ is a continuous function for all real values of $x$ , then $f(8)$ is equal to __20__.

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$f(x)=\begin{cases} 4-x, & x\leq 2 \\ kx-4, & x> 2 \end{cases}$
Probability density function=1
$\int_{-\infty}^{+\infty} f(x)dx=1$
$\int_{-\infty}^{2} f(x) dx+\int_{2}^{+\infty} f(x) dx=1$
$\left [ (4x-\frac{x^2}{2}) \right ]_{-\infty}^{2}+\left [ (k\frac{x^2}{2}-4x) \right ]_{2}^{+\infty}=1$
$[(8-2)-0]+[0-2k+8]=1$
6-2k+8=1
-2k=-13
k= $\frac{13}{2}$
Now,
f(x)=kx-4
$f(8)=(\frac{13}{2} \times 8-4)$
f(8)=52-4
f(8)=48

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