Answer: [tex]g(x) = \frac{1}{x-6} - 4[/tex]
This can be written out as g(x) = 1/(x-6) - 4
The "x-6" is in the denominator, while the "-4" is not.
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Explanation:
We'll start with the parent function [tex]f(x) = \frac{1}{x}[/tex]
When we shift 4 units down, we're subtracting 4 from the y coordinate of each point on the curve. Which is the same as subtracting 4 from f(x) because y = f(x).
So we have [tex]h(x) = f(x) - 4 = \frac{1}{x} - 4[/tex] as an intermediate step.
Then to shift 6 units to the right, we'll replace every x with "x-6". Imagine we kept the h(x) curve completely still, and instead we moved the xy axis 6 units to the left. This would give the illusion of h(x) moving 6 units to the right if we made the xy axis stay still. So that's why we go for "x-6" instead of "x+6".
Therefore, we end up with [tex]g(x) = \frac{1}{x-6} - 4[/tex]
Side note: plugging in x = 6 leads to a division by zero error. This would mean x = 6 is not in the domain.
