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Numerade Educator



Problem 1 Medium Difficulty

Use the given graph of $ f $ to find a number $ \delta $ such that
if $ \left| x - 1 \right| < \delta $ then $ \left| f(x) - 1 \right| < 0.2 $



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Video Transcript

in this problem. Were given a graph of F. And our graph of F looks something like this is X and Y. And we have a curve And here's one going up here over to here. And yes, up here, down to here, 0.7. This is 1.2 and under here Is 1.1 And this is 0.8 mm. And we're asked to find a number delta such that what's that? What If AB Survive X -1 is less than delta, then F of X -1. Absolute value Is less than 0.2. Okay, So delta is going to be the distance. We are from one either positive or negative distance. Right? So that X -1 is less than that distance. Okay, So from the graph from the data we have here, right, we can see that We're we've got a delta, either .3 or of one. Delta is .3 or .1 right, So the f x is one of those points, Then f of X must one is less than .2. But here's the problem. If I choose .3, then that means that I get to choose. And that means I can have X equals 1.3, Which means then that F of X -1, It's now greater than .2, isn't it? Because what will happen, I will end up with a point out here At 1.3. And if that curve continues on, I'm way down here. And so then this distance here is greater And 0.2. So what does that tell me? It tells me that only delta Equal to 0.1 works for this problem, so that's my number delta such that F of X -1 is always less than .2.