Question

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

Answer 1

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, ∞ )`

Explanation:

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]}