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Suppose $f$ is a periodic function. The period of $f$ is 5 and $f(1)=2 .$ Find $f(6),$ $f(11),$ and $f(-4) .$...

Question

Suppose $f$ is a periodic function. The period of $f$ is 5 and $f(1)=2 .$ Find $f(6),$ $f(11),$ and $f(-4) .$

Suppose $f$ is a periodic function. The period of $f$ is 5 and $f(1)=2 .$ Find $f(6),$ $f(11),$ and $f(-4) .$



Answers

Suppose $f$ is a periodic function. The period of $f$ is 5 and $f(1)=2 .$ Find $f(6),$ $f(11),$ and $f(-4) .$

Asked to determine if the falling functions are periodic and if so, to find the period. So for a C the graph over here, this is periodic. The reason why is there's the same repeating pattern time after time. So yes, it's periodic. And the period as you can see, is gonna be too. We have one pattern within to the interval to this one. Upside down V is gonna be our period stands was gonna be too. Okay, Part B, this is not periodic. As you can see, it becomes skinnier and smaller with lower height. So therefore, this is no, this is not periodic, so we can't determine the period. It is not consistent for Graf. See, this is also not periodic. The reason why is it looks periodic up until it gets to the origin. Then you can see that these to form a sort of em like this, where they meet up at the origin. However, this is not the same consistent period. Oh, it's not the same consistent pattern all the way from negative 44. It deviates slightly towards the origin. So this is not periodic. No for D. As you can see this is periodic. It is the same repeating pattern time after time. And we know that our period for this would be three. Because if you look at the graph, you can see that from here to here. Three. We have one complete period, one complete pattern within three and then up told six. We have another complete potter and up until 9 11 complete powder.

Okay, so this question wants us to consider how periodic functions differ in rectangular coordinates versus polar coordinates. So first, let's consider the case or more familiar with in rectangular so it says have a period of two pie. So the classic function we use here is why equals sign of X? So this is periodic on to pie. So what happens with Sign of X? So let's graph. So sign of ex starts at zero goes up to one suspect. Zero goes down the negative one, and then it completes its period. But now we'll use a different color for its next period, and we see that as this keeps repeating through its periods, the graph is literally just repeating. The only thing different about the graph for each of the three periods is the color. So this zero to pie. This is too pie for pie, and this is for pi to six by so for rectangular coordinates, periodic functions draw out the same pattern every period, and that should make sense because since we only have a limited range of fly values, every time they cycled through, it hits them all again. So now let's consider the more interesting case of ah, polar graph that's periodic. So let's consider our equals co sign data. Actually, let's do sign data just to stay consistent so we can show really how different these two are, even though it's the same function, just a different coordinate system. So science data is a circle centered at positive 1/2 on the Y axis. And if we think about this every time that sign theater repeats, it hits the same R value as it did the last period. So that means that we're hitting the same distance from the origin over and over again. So let this be one period right here, for example. The whole graph is traced out in one period. So that's why we usually restrict our domain. Because if we don't Mike, let's say we graph for another period. It just traces over itself, and this will keep going. The graph just keeps repeating and tracing over itself. So we really gained nothing new from graphing these additional coordinates. So what can we say for polar graphs? They trace over themselves every period. So now let's look at the difference. So for rectangular graphs were drawing the function again, but it's X coordinate is different. But this time, when we draw the function again, its distance from the origin never changes, so we're just drawing over it again.

All rights. We have the equation f of A is equal to to a plus three and were asked to find ah, bunch of different values for a So the 1st 1 we've got here is half of negative five, so I just plug in where there is an A I plug in negative five. So two times a negative five is negative. 10 plus three is equal to negative seven. Then we're asked to plug in F of negative three. So where there's an A, I plug in a negative three and I get a negative six year, two times a negative three is the negative. Six plus three is a negative three, then were asked for half of 1/2. So I plug in a 1/2 where there's an a plus three. Two times of 1/2 is one plus three is four, and then a finally and half of four and 1/2 before is two times of four plus three. So two times four is eight plus three is 11

Special wanna find f of negative thoughts. So we're just gonna plug whatever numbers inside of these parentheses here in for that variable. So for this first example, we're gonna plug in negative vibe, and we were the way, see, X and our function, they just become. Now, you make sure you bring down a negative, an insert negative five for X and then bring down it. Negative. Seven. There. So think of this as, ah, negative. One times negative fives and understood. One made one times negative. Bach. Is this positive? Five minus seven in five months. Seven. Gives us negative, too through. Gonna do the same thing for every negative three. We're gonna plug negative three and bricks. So remember, understood. Negative point. Now we're gonna plug in negative three for a variable x and bring down that one. A seven negative one times negative. Three gives us positive three. Then three months. Seven gives us negative for. We also want to find after 1/2. So negative 1/2 numbering down at minus seven. This would be negative 1/2 minus seven. And when it's due common denominators so negative. 1/2 minus. And we need this also have denominator too. So this hasn't understood denominator one. So we multiply the top and bottom by two. And we get negative 14 on top. So one minus 14. That gives us negative 15 over, too. Or if you like my extractions. Negative. Seven and 1/2. And lastly, we need to find ever before stood up before it was gonna plug in or everywhere that we see x so negative or minus seven was becomes negative for money. Seven. And that gives us negative 11.


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