To find and simplify [tex]\(\left(\frac{f}{g}\right)(-2)\)[/tex] given the functions:
[tex]\[ f(x) = x(-2x - 3) \][/tex]
and
[tex]\[ g(x) = -2x - 3 \][/tex]
we need to follow these steps:
1. Compute [tex]\( f(-2) \)[/tex]:
[tex]\[ f(x) = x(-2x - 3) \][/tex]
Substitute [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = (-2)(-2(-2) - 3) \][/tex]
Simplify inside the parentheses:
[tex]\[ -2(-2(-2) - 3) = -2(4 - 3) = -2 \cdot 1 \][/tex]
So,
[tex]\[ f(-2) = -2 \][/tex]
2. Compute [tex]\( g(-2) \)[/tex]:
[tex]\[ g(x) = -2x - 3 \][/tex]
Substitute [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = -2(-2) - 3 \][/tex]
Simplify the expression:
[tex]\[ -2(-2) - 3 = 4 - 3 = 1 \][/tex]
So,
[tex]\[ g(-2) = 1 \][/tex]
3. Compute [tex]\(\left(\frac{f}{g}\right)(-2)\)[/tex]:
[tex]\[ \left(\frac{f}{g}\right)(-2) = \frac{f(-2)}{g(-2)} \][/tex]
We already have:
[tex]\[ f(-2) = -2 \][/tex]
[tex]\[ g(-2) = 1 \][/tex]
So,
[tex]\[ \left(\frac{f}{g}\right)(-2) = \frac{-2}{1} = -2 \][/tex]
Therefore, [tex]\(\left(\frac{f}{g}\right)(-2)\)[/tex] is [tex]\(-2\)[/tex].
