To find \((h - g)(-2)\), we need to evaluate \(h(-2)\) and \(g(-2)\) individually, and then subtract \(g(-2)\) from \(h(-2)\).
1. Evaluate \(h(-2)\):
Given the function \(h(x) = x + 6\), substitute \(x = -2\):
[tex]\[
h(-2) = -2 + 6 = 4
\][/tex]
2. Evaluate \(g(-2)\):
Given the function \(g(x) = \sqrt{x - 5}\), substitute \(x = -2\):
[tex]\[
g(-2) = \sqrt{-2 - 5} = \sqrt{-7}
\][/tex]
The square root of a negative number introduces an imaginary component. Therefore,
[tex]\[
\sqrt{-7} = i\sqrt{7}
\][/tex]
where \(i\) is the imaginary unit.
3. Combine the results:
We have \(h(-2) = 4\) and \(g(-2) = i\sqrt{7}\). Now, we find \((h - g)(-2)\) by subtracting \(g(-2)\) from \(h(-2)\):
[tex]\[
(h - g)(-2) = h(-2) - g(-2) = 4 - i\sqrt{7}
\][/tex]
The result is a complex number, \(4 - i\sqrt{7}\), which is not simply an integer or a fraction. Therefore, the correct choice is:
[tex]\[
\boxed{\text{A }}
\][/tex]
And fill in the box with:
[tex]\[
4 - i\sqrt{7}
\][/tex]
