Find the Domain of the function step by step {√(4-x)+√(3+x)}÷{x^2-1}

Question

Find the Domain of the function step by step {√(4-x)+√(3+x)}÷{x^2-1}

Answer 1

I'm assuming the question is asking about a real valued function, i.e. you want the domain to exclude values which give complex image.

First thing you'd want to do is to find where the function is undefined, or where the denominator is 0. Set the denominator equal to 0 and you find that x² cannot equal 1, so x cannot be +1 or -1.

Next, find the values of x where the function becomes complex, or the radicand is negative. If x is greater than 4, then 4-x is negative and its square root is complex. If x is less than -3, then x+3 is negative and its square root is complex.

Putting those together, x must be greater than or equal to -3 and less than or equal to 4, but it also can't be 1 or -1. Thus, the domain is [-3,-1) (-1,1), (1,4].