Question

Let `f(x)=x^2` and `g(x)=sqrtx`, how do you find the domain and rules of `(f/g)(x)`?

Answer 1

Domain: `x > 0`
`(f/g)(x)=sqrt(x^3)" or, equivalently "x^(3/2)`

Explanation:

`(f(x))/(g(x))=(x^2)/sqrt(x)` ...and since
`color(white)("XXXXXX")`division by zero is undefined, and
`color(white)("XXXXXX")`square root of negative numbers are not defined for Real values
`color(white)("XXX")`we can eliminate any values of `x <= 0`
leaving us with a domain of `x > 0`

`x^2/sqrt(x)=x^2xxx^(-1/2)=x^(3/2)` or
`color(white)("XXXXXXXXXX")=(sqrt(x))^3` or
`color(white)("XXXXXXXXXX")=sqrt(x^3)^2`