Multivariable calculus

[cancerfromspace]

Hi everyone, I need help with finding domain and range of this function of 2 variables. Thank you so much

Square root of x*ln(y^2 - x)

 

[Ishuda]

Hi everyone, I need help with finding domain and range of this function of 2 variables. Thank you so much

Square root of x*ln(y^2 - x)

Given the function
f(z) = [x ln(y²-x)]^1/2
which maps RxR to R the results of the function must be real. Starting with the argument of the natural log function, this means y² is greater than x. Having satisfied that, we now must also have x ln(y²-x) be non negative or either (a) both x and ln(y²-x) are non negative or (b) both x and ln(y²-x) are non positive. Taking it one step further, since ln(y²-x) is non-negative if (y²-x)\(\displaystyle \le\)e, the latter two conditions can be written as (a) both x and (y²-x -e) are non negative or (b) both x and (y²-x-e) are non positive.

Put all of that together

 

[HallsofIvy]

Or did you mean \(\displaystyle \sqrt{x} ln(y^2- x)\)? It is not clear from "square root of x(y^2- x)" if the square root applies to the whole think or only to the x.

 

[cancerfromspace]

Or did you mean \(\displaystyle \sqrt{x} ln(y^2- x)\)? It is not clear from "square root of x(y^2- x)" if the square root applies to the whole think or only to the x.

it applies to the whole equation