Graph each inverse circular function by hand.
y = arcsec...

Question

Graph each inverse circular function by hand.

y = arcsec [(1/2)x]

Accepted Answer

Hey, everyone in this problem, we're asked to graph the following inverse trigonometric function. So we have Y is equal to C can inverse of a quarter X. Now it can be hard to picture what these inverse functions look like. So what we wanna do is actually think about the inverse of the inverse function that we want a graph. And we know that if we can draw that graph, then this graph, the inverse will just be a reflection across the line Y is equal to X. OK. So let's start by finding the inverse of the function that we're trying to graph, we have Y is equal to C can inverse of a quarter X. What we're gonna do is try to isolate for X in order to find our ins So if we do that, we can take C can on both sides, we get that C can of Y is equal to a quarter X because C can't, if C can't inverse those two are gonna cancel it. And then we can multiply both sides by four, we get that X is equal to four multiplied by C can of Y. And then we just swap X and Y to write this in terms of our normal Y equals function. If we swap X and Y, we get Y is equal to four C can't. So the inverse of the function we're trying to graph is Y is equal to four C can FX. OK. So we're gonna try to graph this function and we know that four C of X is equivalent to four divided by cosine of X. OK. So whenever cosine of X is zero, we're gonna have vertical asymptotes, the denominator is gonna be equal to zero, the function will not exists there. So we're gonna have a bunch of vertical asymptotes. When cosign of X is equal to zero, we know that cosine of X is equal to zero if X is equal to pi divided by two, three pi divided by two, et cetera. And we're gonna put plus or minus here and there are more points where cosine of X is equal to zero, we're just choosing the ones that fall within the X values we have on our graph. OK. So these are the ones that are going to be on our graph and we can actually go up and add those to our graph right now. And so we have these vertical asymptotes negative three pi divided by two pi divided by two, sorry, negative pi divided by two. How about about you? And three pie divided by two? OK. So these are the vertical asymptotes of the inverse. OK. And again, remember that what we're doing is gonna graph the inverse of the function we wanna find. And then we're gonna reflect it across the line Y is equal to X. All right. So let's keep going with this function Y is equal to four C can of X or four divided by cosine of X. We're gonna do a little table of values to get an idea of what this function looks like. So we have our X values and our Y values, we're gonna choose X values of zero and we'll look at plus or minus pi and we can do this because we know cosine is an even function. And so on both sides, we're gonna be getting that same value. So plus or minus pi plus or minus two pi plus or minus pi divided by four plus or minus three pi divided by four A plus or minus five pi divided by four and plus or minus seven pi divided by four. These are the values we want, OK, when X is zero and cosine of zero is just one. So our function is just four. And again, this is the inverse of the function. We actually wanna go, well, we have pie cosine of X is going to be negative one. So we're gonna get negative four at two pi cosine is back to being one. So we get positive four. Again, one, X is pi divided by four plus or minus, we're gonna get that cosine of X is one divided by the square root of two. And if we take four, divide by one, divided by the square root of two, we get four multiplied by the square root of two. Similarly, for three pi divided by four, we get negative four, multiplied by the square root of two. We get that same value negative four root two or plus or minus five pi divided by four. And then we get the positive value again, four root two for plus or minus seven pi divided. OK. So we have an idea of what the inverse of the function we want graph is. So let's go up to our graph and we've written all of these values out, but we actually don't need them. All right, we have to restrict ourselves, we need to find a domain where cosine of X and sequence of X are 1 to 1. OK? They have to be 1 to 1 for this inverse to be valid, right? And so we're gonna restrict ourselves 20 to P if we restrict ourselves to zero to pi and we graf this function four divided by cosine of X at X equals zero, we know we have four and we know that that's increasing. So it's gonna be increasing up to that vertical aceto. And then I, we know we have a value of negative four and as we move to the left towards the vertical Asymptote that's decreasing. So this is our function in black Y is equal to four divided by cosine of X. This is the inverse of the function we want to grow, right? We have the inverse, we restricted it to this domain where that inverse is valid. And now the last thing is to reflect this across the line Y is equal to X. So let's think about this line Y is equal to X. And we're gonna draw this as a dotted line so that it's clear. So we have 00. And when we're at pi we're gonna be at pi on the Y axis as well. When we're at T pie, we're gonna be at TP or just over six until we get this line Y equals X. And that's not a perfect drawing, but we are going to be reflecting about this line and we can do the same down into the negative direction as well. OK. So that's the line we're gonna reflect about. And when we do that and what we're really doing is swapping the X and Y values here, we already swapped the X and Y values algebraically to get the inverse. Now we're kind of going backwards, we have our graph and we're gonna swap the X and Y values in our graph. We look at this top part and we have an X value of zero and A Y value of four. So now we're gonna have a Y value of zero and then X value of four, you know four is gonna be somewhere after pi yes. And we can do the same for those other values. And our function is going to go something like this off to the right. All right. Now we can do the same for this bottom part of our function. OK? We swap the X and Y values. And what we get is that this function is going to be starting around pi OK. We had an X value of pi so now we have a Y value of pi it's gonna go something like this lay down until and so the red is the graph of our function. What that we want to graph? Hey, this is seeking an inverse of a quarter X and we got there by actually graphing the inverse and reflecting it across that line. Why equals X? Thanks everyone for watching. I hope this video helped see you in the next one.

Graph each inverse circular function by hand.
y = arcsec [(1/2)x]

Key Concepts

Inverse Circular Functions

Domain and Range of Inverse Functions

Transformations of Functions

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