What are the domain and range of the real-value function f (x) = -2+√x+7

Because the function f (x) = -2+√x+7 has √x, x should be more or equal 0.

Domain is x≥0, or [0,+∞).

The smallest value of x=0, so the smallest value of f (x) is f (0) = -2+√0+7=5,

so range is f (x) ≥5, or [5, + ∞).

The given function - 2+√x+7 is a square root function.

The expression inside the square root of a unction cannot be negative. Because if it will be negative then the function will be imaginary. So, expression inside the square root must be positive or zero. Hence, to find the domain of a square root function set the expression inside the square root as more than or equal to zero.

So, x+7≥0

x+7 - 7 ≥0-7 Subtract 7 from each sides to isolate x.

x≥ - 7

So, the domain of the given function is x≥ 7.

Notice that if we will plug in any value which is more than or equal to 7then we will get y >=-2 only. It must be - 2+.

So, range of this function is y>=-2.