Let's evaluate the given functions step-by-step at [tex]\( x = -6 \)[/tex].
First, we need to evaluate [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 7x \][/tex]
[tex]\[ f(-6) = 7 \cdot (-6) = -42 \][/tex]
Next, we evaluate [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = \frac{1}{x - 3} \][/tex]
[tex]\[ h(-6) = \frac{1}{-6 - 3} = \frac{1}{-9} = -\frac{1}{9} \][/tex]
Now, we need to find the value of the function [tex]\( \left(\frac{h}{f}\right)(x) \)[/tex]:
[tex]\[ \left(\frac{h}{f}\right)(x) = \frac{h(x)}{f(x)} \][/tex]
At [tex]\( x = -6 \)[/tex]:
[tex]\[ \left(\frac{h}{f}\right)(-6) = \frac{h(-6)}{f(-6)} = \frac{-\frac{1}{9}}{-42} \][/tex]
Simplify the expression:
[tex]\[ \left(\frac{h}{f}\right)(-6) = \frac{-\frac{1}{9}}{-42} = \frac{1}{9 \cdot 42} = \frac{1}{378} \][/tex]
Hence,
[tex]\[ \left(\frac{h}{f}\right)(-6) = \frac{1}{378} \][/tex]
This fraction simplifies to a decimal:
[tex]\[ \frac{1}{378} \approx 0.0026455026455026454 \][/tex]
So the final answer is:
[tex]\[ \left(\frac{h}{f}\right)(-6) \approx 0.0026455026455026454 \][/tex]
Therefore, the value of [tex]\( \left(\frac{h}{f}\right)(-6) \)[/tex] is [tex]\( 0.0026455026455026454 \)[/tex].
