Question

How do you find the domain of `f(x) =1/sqrt(-x^2+5x-4)`?

Answer 1

The domain is `x in (1,4)`

Explanation:

For the square root and the division, the conditions are

`(-x^2+5x-4)>0` and `x!=1` and `x!=4`

Let `f(x)=(-x^2+5x-4)=-(x^2-5x+4)=-(x-1)(x-4)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaaa)` `-oo` `color(white)(aaaaaa)` `1` `color(white)(aaaaaaaa)` `4` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `-(x-1)` `color(white)(aaaaa)` `+` `color(white)(aaa)` `||` `color(white)(aaa)` `-` `color(white)(aaaaaa)` `-`

`color(white)(aaaa)` `(x-4)` `color(white)(aaaaaaa)` `-` `color(white)(aaa)` #color(white)(aaaa)-# `color(white)(aa)` `||` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaaa)` `-` `color(white)(aa)` `||` `color(white)(aaa)` `+` `color(white)(aa)` `||` `color(white)(aaa)` `-`

Therefore,

`f(x)>0`, when `x in (1,4)`

graph{1/sqrt(-x^2+5x-4) [-9.55, 18.93, -4.48, 9.76]}