Question

Let $f(x)=\frac{1}{x}, g(x)=\frac{1}{4 x^2-1}$ and $h(x)=\frac{5 x}{x+2}$ be three functions and $k(x)=h(g(f(x)))$. If domain and range of $k(x)$ are $R-\left\{a_1, a_2, a_3, \ldots . a_n\right\}$ and $R-A$ respectively where 'R' is the set of real numbers then

• A. $ n +\displaystyle\sum_{ i =1}^{ n } a _{ i }=5$
• B. $n +\displaystyle\sum_{ i =1}^{ n } a _{ i }=10$
• C. number of integers in set A is 5
• D. Number of integers in set $A$ is 7.

Answer 1

Answer: (A), (D)

$k ( x )= h ( g ( f ( x )))=\frac{5 x ^2}{8- x ^2} $
$\text { Domain of } k ( x ) \text { is } R -\{0, \pm 2, \pm 2 \sqrt{2}\} $
$\text { Range of } k ( x ) \text { is }(-\infty,-5) \cup(0, \infty)-\{5\}$
$ \text { or } R -([-5,0] \cup\{5\}) $
$\therefore A \text { is }[5,0] \cup\{5\} $