What are the domain and range of the real-valued function f(x)=−1+√x+6𝑓(𝑥)=-1+𝑥+6Domain:  Range:

The function you've given seems to be a bit confused, as it's written in two different ways. However, I'll assume you're asking about the function f(x) = -1 + √(x+6).

The domain of a function is the set of all possible input values (x-values) which will output real numbers. For this function, the domain is determined by the restriction of the square root function. Since we can't take the square root of a negative number (in the real number system), the value inside the square root (x+6) must be greater than or equal to 0. So, solving the inequality x+6 >= 0, we get x >= -6. Therefore, the domain of the function is x >= -6 or in interval notation, [-6, ∞).

The range of a function is the set of all possible output values (y-values). For this function, since the square root function outputs values of 0 or greater and we're subtracting 1, the smallest possible value is -1. Therefore, the range of the function is y >= -1 or in interval notation, [-1, ∞).