Exploring the Inverse of Daylight Hours as a Function of the Day of the Year

[Imperil]

The following graph shows the number of daylight hours as a function of the day of the year. Draw the inverse of the graph, and describe what the inverse represents.

The graph given is very close to this (which I believe is close enough for this question): http://www.math.unl.edu/~bharbour/M106/projects/grphI1.gif

I believe the inverse of this function is basically this graph rotated 45 degrees to the right, as if you plot the inverse points (i.e. (8, 10) instead of (10, 8), etc you end up with this. Am I correct in thinking this?

My problem is that I am not sure what the inverse represents as it doesn't make a lot of sense to me (which is one reason I think I may have the inverse totally wrong). I think the inverse would no longer be a function because many pairs have multiple Y values per X. Also the graph gives odd results such as (135 days have 24 hours of daylight), so I am obviously missing some concept.

Thanks for any help.

 

[Mark44]

The following graph shows the number of daylight hours as a function of the day of the year. Draw the inverse of the graph, and describe what the inverse represents.

The graph given is very close to this (which I believe is close enough for this question): http://www.math.unl.edu/~bharbour/M106/projects/grphI1.gif

I believe the inverse of this function is basically this graph rotated 45 degrees to the right,

No.

as if you plot the inverse points (i.e. (8, 10) instead of (10, 8), etc you end up with this.

Sort of. You plot the pairs of numbers in opposite order. If (8, 10) is on the original graph, (10, 8) will be on the graph of the inverse function/relation. Pick a bunch of points on the original graph, and graph the reversed ordered pairs. What you end up with is not the original graph being rotated by 45 degrees to the right (or left); the new graph is the reflection across the line y = x.

Since the graph you show is not one-to-one, the inverse you get will not be a function. For a given value on the horizontal axis, there will be multiple values on the vertical axis.

Th

Am I correct in thinking this?

My problem is that I am not sure what the inverse represents as it doesn't make a lot of sense to me (which is one reason I think I may have the inverse totally wrong). I think the inverse would no longer be a function because many pairs have multiple Y values per X. Also the graph gives odd results such as (135 days have 24 hours of daylight), so I am obviously missing some concept.

Thanks for any help.

As you mentioned, the graph you show gives the number of daylight hours as a function of the date. For most dates, there will be two (maybe three) dates with the same number of daylight hours. If you specify a date, you can read off the number of daylight hours.

The inverse relation gives the relationship in the opposite order. If you specify the number of hours of daylight, you can read off the date(s). In the inverse graph, because you reflected the points across the line y = x, the labels on the axes got reflected as well, so the inverse relation has hours of daylight on the hor. axis, and dates on the vertical axis.

If your graph is showing that 135 days have 24 hours of daylight, yes, you are definitely doing something wrong.

 

[Imperil]

Thank you so much for the help :) Sorry my wording was really bad, and what I did was plot the pairs in opposite order which resulted in a graph that looked rotated 90 degrees, although I shouldn't have said that. The graph I drew is definitely a reflection along the Y axis. (hence the reason for saying it was rotated 90 degrees)

The big problem I had was that I forgot to reflect the labels! Now that I have done that the graph makes complete sense, if I rotate my paper it is the same as the original graph (although with the axis on the right obviously). As I said the problem I have has a graph that is slightly different than the one I posted, I should have scanned it instead. It starts at (0, 8) and ends at (360, 8) which is why the inverse just looks rotated 90 degrees.

Thanks so much again!

 

[Mark44]

The graph I drew is definitely a reflection along the Y axis. (hence the reason for saying it was rotated 90 degrees)

No, it's not a reflection across the Y axis, which would move a point (a, b) to (-a, b). This is not a rotation by any amount. Did you mean "reflection cross the line y = x"?

Let me say it again: You can't get a graph of the inverse function/relation by rotation. You can do this, however, if your graph is on paper that is thin enough to see through. Pick the paper up, holding it by the upper right and lower left corners. Turn it over. What was the horizontal axis should now be pointing up and down, and what was the vertical axis should now be pointing left and right. The graph on the backside of the paper should now appear through the paper as the inverse.

 

[Imperil]

Mark thanks so much for your help, I actually did have it correct it just turns out I am very bad with the terminology so I'll definitely have to improve on that. I turned over the paper and checked it, it turned out correct so I was happy about that.

Thanks again!