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Q. Let $f, g \& h$ be three functions defined as follows :
$f(x)=\frac{32}{4+x^2+x^4}, g(x)=9+x^2 \quad \text { and } \quad h(x)=-x^2-3 x+k$ Identify which of the following statement(s) is(are) correct?

Relations and Functions - Part 2

Solution:

$f ( x )=\frac{32}{4+ x ^2+ x ^4} ; g ( x )=9+ x ^2 ; h ( x )=- x ^2-3 x + k$
(A) range of $f$ is $(0,8]$
(B) $ h ( f ( x ))>0$ and $h ( g ( x ))<0$
$h (0) \geq 0 \Rightarrow k \geq 0 $
$h (8)>0 \Rightarrow-64-24+ k >0 \Rightarrow k >88$
$h (9)<0 \Rightarrow-81-27+ k <0 \Rightarrow k <108$
Number of integral values of $k$ is 19 .
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(D) Maximum value of $g(f(x)) is g(8) = 64 + 9 = 73$.