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Problem 30 Hard Difficulty

Use the given graph of $ f $ to sketch the graph of $ y = \frac{1}{f(x)} $. Which features of $ f $ are the most important in sketching $ y = \frac{1}{f(x)} $? Explain how they are used.

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okay, we have the graph of y equals F of X, and we want to graph y equals the reciprocal of f of x one over f of X. So the most important features here are going to be the X intercepts of F of X. Because those have a y coordinative zero. And what happens when you take one and divided by zero? It's undefined. So those will be the places where the reciprocal function has vertical Assen totes, so we'll put those in place. Another point that we could use is on the original f of X. We have the 0.0.1 So let's take the reciprocal of one. We get one, and so we still have the 10.1 Okay, now let's figure out what to do beyond that. So looking at this region right here, we have negative numbers getting further from zero their reciprocal czar going to be negative numbers, getting closer to zero. And then let's look at this section in here those air positive numbers and as we move out farther from X equals zero toward the sides, we have positive numbers getting closer to zero. Their reciprocal czar going to be positive. Numbers getting farther from zero. Okay, let's take a look at this section right here. Those white coordinates are negative numbers, and as we move to the sides, they're getting closer to zero. So we're going to have why? Coordinates that are negative. Getting farther from zero. And then this section right here with positive numbers getting farther from zero. So they're reciprocal, Czar. Going to be positive numbers, getting closer to zero. Now, let's do the first section again. Okay? So, as I said, those were negative numbers that get farther from zero. The further we go to the left. So let me fix that. So these were going to be negative numbers. They get closer to zero, the further we go to the left.