Let [tex]$h(x)=x+6[tex]$[/tex] and [tex]$[/tex]g(x)=\sqrt{x-5}$[/tex].

Find [tex]$(h - g)(-2)$[/tex].

Select the correct choice below and fill in any answer boxes within your choice:

A. [tex]$(h - g)(-2)=$ \boxed{\text{ }}[/tex] (Simplify your answer. Type an integer or a fraction.)

B. The function value does not exist.



Answer :

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]