State the domain of the function: g(x) = Squareroot(5 + 8x)

Answer 1

The domain of a function g(x) (unless otherwise specified) is the set of x values that we can plug into the function (loosely speaking). In the case of square roots we know that we cannot take the square root of a negative number. So, in our case here we need to make sure that 5+8x remains ≥ 0. Subtract 5 and then divide by 8 on both sides of the inequality to see that we require x ≥ -5/8. So the domain in this case is x ≥ -5/8 which you might also see written as (-5/8,∞).

A few quick notes:

You will be given problems like this with different types of functions other than just square roots. Different functions have different domain restrictions to be on the lookout for. The most common ones you will likely encounter include functions which will have you dividing by zero (which is mathematically undefined). For example, for f(x) = 5/(x-3) the domain is all values x except for x = 3, because when we plug in 3 we will be dividing by zero. For some functions (polynomials are a common example) the domain is all real numbers because whatever number we choose to plug in will output a real number.