To evaluate [tex]\(\left(\frac{g}{f}\right)(3)\)[/tex], we need to do the following steps:
1. Evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 3\)[/tex].
2. Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 3\)[/tex].
3. Calculate [tex]\(\frac{g(3)}{f(3)}\)[/tex] and determine if we have a division by zero.
First, let's evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 3\)[/tex]:
[tex]\[ g(x) = |x - 3| \][/tex]
[tex]\[ g(3) = |3 - 3| = |0| = 0 \][/tex]
Next, let's evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 3\)[/tex]:
[tex]\[ f(x) = -6x \][/tex]
[tex]\[ f(3) = -6 \cdot 3 = -18 \][/tex]
Now, let's calculate [tex]\(\frac{g(3)}{f(3)}\)[/tex]:
[tex]\[ \frac{g(3)}{f(3)} = \frac{0}{-18} = 0 \][/tex]
Since we are not dividing by zero in this step, the calculation is valid. So, the value of [tex]\(\left(\frac{g}{f}\right)(3)\)[/tex] is:
[tex]\[ \left(\frac{g}{f}\right)(3) = 0 \][/tex]
Therefore, [tex]\(\left(\frac{g}{f}\right)(3)\)[/tex] is [tex]\(\boxed{0}\)[/tex].
