Domain [-5,∞)
Range [0,∞)
Step-by-step explanation:
Part 1) Find the domain
we have
![f(x)=\frac{1}{2}\sqrt{x+5}](/tpl/images/0478/6138/434e0.png)
we know that
The radicand must be greater than or equal to zero
so
![x+5\geq 0](/tpl/images/0478/6138/6d181.png)
solve for x
subtract 5 both sides
![x\geq -5](/tpl/images/0478/6138/40b17.png)
The solution for x is the interval [-5,∞)
All real numbers greater than or equal to -5
Remember that
The domain of a function is the set of all possible values of x
therefore
The domain of the function f(x) is the interval [-5,∞)
Part 2) Find the range
we have
![f(x)=\frac{1}{2}\sqrt{x+5}](/tpl/images/0478/6138/434e0.png)
Find the value of f(x) for the minimum value of x
For x=-5
![f(x)=\frac{1}{2}\sqrt{-5+5}](/tpl/images/0478/6138/32024.png)
![f(x)=0](/tpl/images/0478/6138/ce73d.png)
The minimum value of f(x) is equal to zero
so
The solution for f(x) is the interval [0,∞)
All real numbers greater than or equal to 0
Remember that
The range of a function is the complete set of all possible resulting values of y, after we have substituted the domain.
therefore
The range of the function is the interval [0,∞)