Question

How do you find the domain and range of `f(x) = - sqrt2 / (x² - 16)`?

Answer 1

Domain is all value of `x` other than `-4` and `4`.

Range is all values except those between `0` and `sqrt2/16` i.e. `(0,sqrt2/16)`

Explanation:

AS `f(x)=-sqrt2/(x^2-16)=-sqrt2/((x+4)(x-4))`,

it is quite apparent that `x` cannot take values `-4` and `4`.

Hence domain of `x` is all values other than `-4` and `4`.

Further, when `x` lies between `-4` and `4` i.e. `|x|<4`,

the denominator is always negative i.e. `f(x)` is positive and maximum value of denominator is when `x=0`, and then `f(x)=sqrt2/16`. Hence this is minimum value of `f(x)` in this range.

When `|x|>4`, `f(x)` is always negative and when `x->oo`,

`f(x)=-sqrt2/(x^2-16)=-(sqrt2/x^2)/(1-16/x^2)` i.e. `f(x)->0`

Hence range of `f(x)` is all values except those between `0` and `sqrt2/16`

graph{-sqrt2/(x^2-16) [-5, 5, -2.5, 2.5]}