Domain of the function f(x) = [ _ 10 ( 5x - x^2 4 ) ]^ 1/2 is - Vidyadip

MCQ

Domain of the function $f(x) = {\left[ {{{\log }_{10}}\left( {\frac{{5x - {x^2}}}{4}} \right)} \right]^{1/2}}$ is

A
$ - \infty < x < \infty $
✓
$1 \le x \le 4$
C
$4 \le x \le 16$
D
$ - 1 \le x \le 1$

✓

Answer

Correct option: B.

$1 \le x \le 4$

b
(b) We have $f(x) = {\left[ {{{\log }_{10}}\left( {\frac{{5x - {x^2}}}{4}} \right)} \right]^{1/2}}$…..(i)

From (i), clearly $f(x)$ is defined for those values of $x$ for which ${\log _{10}}\left[ {\frac{{5x - {x^2}}}{4}} \right] \ge 0$

==> $\left( {\frac{{5x - {x^2}}}{4}} \right) \ge {10^0} \Rightarrow \left( {\frac{{5x - {x^2}}}{4}} \right) \ge 1$

==> ${x^2} - 5x + 4 \le 0$ ==> $(x - 1)(x - 4) \le 0$

Hence domain of the function is $[1, 4].$

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