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

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.

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is