Question: Find the values p(3), p(9), p(11) and p(0) for the function, if they exist.
p(z) = \sqrt{2z - 18}
Find p(3). Select the correct choice and, if necessary, fill in the answer box within your choice.
A. p(3) = [ ]
(Simplify your answer. Type an exact answer using radicals as needed.)
B. The function value does not exist. The square root does not have a real number solution.
Find p(9). Select the correct choice and, if necessary, fill in the answer box within your choice.
A. p(9) = [ ]
(Simplify your answer. Type an exact answer using radicals as needed.)
B. The function value does not exist. The square root does not have a real number solution.
1. **State the problem:** We need to find the values of the function $$p(z) = \sqrt{2z - 18}$$ at $$z=3$$, $$z=9$$, $$z=11$$, and $$z=0$$ if they exist.
2. **Recall the domain rule for square roots:** The expression inside the square root must be greater than or equal to zero for the function to have real values. So, $$2z - 18 \geq 0$$.
3. **Find the domain:**
$$2z - 18 \geq 0$$
$$2z \geq 18$$
$$z \geq 9$$
This means the function is defined only for $$z \geq 9$$.
4. **Evaluate $$p(3)$$:**
Since $$3 < 9$$, the function is not defined at $$z=3$$.
5. **Evaluate $$p(9)$$:**
$$p(9) = \sqrt{2(9) - 18} = \sqrt{18 - 18} = \sqrt{0} = 0$$
6. **Evaluate $$p(11)$$:**
$$p(11) = \sqrt{2(11) - 18} = \sqrt{22 - 18} = \sqrt{4} = 2$$
7. **Evaluate $$p(0)$$:**
Since $$0 < 9$$, the function is not defined at $$z=0$$.
**Final answers:**
- $$p(3)$$ does not exist (square root of negative number).
- $$p(9) = 0$$
- $$p(11) = 2$$
- $$p(0)$$ does not exist (square root of negative number).