Suppose f(x) lies in the interval (2, 6). What is the smallest...

Question

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says if HFZ is a function whose output lies strictly between 10 and 20, what is the minimum value of theta such that the absolute value of H of Z minus 15 is less than theta? We're given 4 possible choices as our answers. For choice A, we have 5, for choice B, we have 6. Choice C, we have 15, and for choice D, we have 10. Now this question wants us to minimize the value of theta such that the absolute value of HFZ minus 15 is less than theta, and we're told that HFZ has a is a function whose output lies strictly between 10 and 20, so the output is on the open interval of 10 to 20. So our endpoints there are not included in this interval. Now, in order to minimize the value of data, we need to maximize the value of the absolute value of H of Z minus 15. So, we can do that by substituting in our end points into this function and see which one gives us the greater result. And so we'll start off with their end point of 10, so we'll have the absolute value of 10 minus 15. Which is equal to 5. We'll do the same thing with the other end point. So that'll be the absolute value of 20 minus 15. is equal to 5. So in this case, we get the same result regardless of what endpoint we use. However, if we did have different results, we'd want to choose the one that is larger. So now that we have our left hand side maximized, we can now look at our value for data. Now, in order for data for our inequality to hold true, that means that data has to be greater than or equal to 5. And here we use greater than or equal to, since our endpoints are not included in the interval of our output. Now, we want to minimize our our value of data. So, to minimize this, we're going to take the minimum value that data can be. In this case, that means data is going to be equal to 5. So that is the answer that I'm actually looking for, and this actually corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

Key Concepts

Absolute Value Inequality

Interval Notation

Finding ε

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