What is the domain and range of h(x)=3x+1/x ?

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Answer 1 · 2018-04-18T11:06:16

Let $y=3x+1/x$

Domain of the function : $D_h : x$ $ϵ$ $R-{0}$

Range of the function : $R_h: y$ $ϵ$ $(- ∞ ,-2sqrt3]∪[2sqrt3, ∞ )$

The function is not defined for $x=0$ .

$:.$ The Domain of the function : $D_h : x$ $ϵ$ $R-{0}$

Y-axis is the vertical asymptote for this curve.

The function has a local maximum at $x=-1/sqrt3$ and a local minimum at $x=1/sqrt3$

To find the minimum positive value attained by this function , take $x>0$ and apply AM-GM inequality on $3x$ and $1/x$ :-

${3x+1/x}/2>=sqrt(3x.1/x)$

$rArr3x+1/x>=2sqrt3$

$:.y>=2sqrt3$ for $x>0$

Similarly when $x<0$ ; then $y<=-2sqrt3$

Thus the range of the function is :

$R_h: y$ $ϵ$ $(- ∞ ,-2sqrt3]∪[2sqrt3, ∞ )$

The graph of the function is given below :-

graph{3x+1/x [-22.8, 22.8, -11.4, 11.4]}