VCE Methods Question Thread!

[SE_JM]

Hello!
I have a question about graphing this graph and its inverse: f(x)= 5*e^2x -3.
I found the inverse of this, which was: f^-1(x)= (1/2)*log_e((x+3)/5)
When i graph both of them on the same axes, they don't intersect.

Note: the x-axes have a 1-interval (-2,-1,0,1,2...)  and the y-axis have a 5-interval (-10, -5, 0, 5 , 10...)

But when i check the answers. they have two points of intersection.
Why is this?
 I checked everything again so I don't think it's my calculations at fault...
Is it because i set my axes intervals differently? (normally, it's not a problem though...)

I'm quite confused...
any help?
Thanks

[keltingmeith]

Hey guys,
I have a question regarding the exponential graph

The question asks:
The number of people, N, who have a particular disease at time t years is given by
N=N₀e^kt
If the number initially is 20000 and the number decreases by 20% each year, find the value of k

I thought k= -0.2 (20/100) but the textbook says it's -0.223

Why? Shouldn't it exactly be -0.2?

Okay, so first we do the easy part - find , which is 20,000, and can easily be seen by substituting N=20,000, t=0.

So, we have equation . Every year, the number DECREASES by 20% - so, in any one year, the population is 80% of the population the year before. So, let's consider one year later:

[keltingmeith]

Hello!
I have a question about graphing this graph and its inverse: f(x)= 5*e^2x -3.
I found the inverse of this, which was: f^-1(x)= (1/2)*log_e((x+3)/5)
When i graph both of them on the same axes, they don't intersect.

Note: the x-axes have a 1-interval (-2,-1,0,1,2...)  and the y-axis have a 5-interval (-10, -5, 0, 5 , 10...)

But when i check the answers. they have two points of intersection.
Why is this?
 I checked everything again so I don't think it's my calculations at fault...
Is it because i set my axes intervals differently? (normally, it's not a problem though...)

I'm quite confused...
any help?
Thanks

Wolfram says there are two solutions, too. I'm guessing you've got something wrong happening with your calculator. Your increments should include those two points, so maybe check you entered them correctly.

[SE_JM]

Wolfram says there are two solutions, too. I'm guessing you've got something wrong happening with your calculator. Your increments should include those two points, so maybe check you entered them correctly.

What's an increment.
Excuse my lack of mathematical terms

[keltingmeith]

What's an increment.
Excuse my lack of mathematical terms

Hahah, I don't actually think it's a mathematical term... It's the interval that your axis indicate it's moving by. For example, on a 30cm ruler, it often has marked increments of 1cm. Your x-axis is moving in 1 unit increments, and your y-axis in 5 unit increments.

[SE_JM]

Hahah, I don't actually think it's a mathematical term... It's the interval that your axis indicate it's moving by. For example, on a 30cm ruler, it often has marked increments of 1cm. Your x-axis is moving in 1 unit increments, and your y-axis in 5 unit increments.

Thank you!
Are we expected to solve for the intersection points by hand in methods course? Can i just use the CAS?

Were i meant to test if the intersection points occur before plotting?

What i did was, i figured out y-int and x-int for both graphs, and i just graphed it without knowing whether intersection happened or not..silly me

Also, does it matter whether the 'increment' (see how fast i'm using this new word? haha) is consistent in the x-axis and y-axis? so for example, if my x-axis have a 5 unit increment and my y-axis has a 2 unit increment does the graph go funny?

The reason why i'm asking is because i found when i was graphing these graphs (and other ones like hyperbolas) sometimes if the increments are different, the graph looks really weird...

Thanks (i ask a lot of question)

[keltingmeith]

Thank you!
Are we expected to solve for the intersection points by hand in methods course? Can i just use the CAS?

Man, I'm not expected to do it in second year university. CAS it, definitely CAS it.

Were i meant to test if the intersection points occur before plotting?

That would definitely be a good idea. Helps with the shape.

Also, does it matter whether the 'increment' (see how fast i'm using this new word? haha) is consistent in the x-axis and y-axis? so for example, if my x-axis have a 5 unit increment and my y-axis has a 2 unit increment does the graph go funny?

Well, remember that inverses are symmetrical along the line y=x. So, if you have a consistent scale across both axis, you can just draw one by sight without having to worry too much about shape (as long as it looks symmetrical).

For an example of this, try drawing the following graphs with both axis set to -10 min and +10 max: (you may also consider including the line y=x)

y=x^2 (x>0) and y=sqrt(x)
y=e^x and y=ln(x)
y=1/(x-4) (x>4)and y=1/x + 4 (x>0)
(the domains are added just so it's easier to look at)

[catherine_mimi]

I was looking at the chapter summary where it said that  b^(log_ba)x is equal to a^x
How is this so? Can you just bring it down to log_bax * b despite it is an exponential?
Thanks,

[catherine_mimi]

Okay, so first we do the easy part - find , which is 20,000, and can easily be seen by substituting N=20,000, t=0.

So, we have equation . Every year, the number DECREASES by 20% - so, in any one year, the population is 80% of the population the year before. So, let's consider one year later:

Thanks for this!

[SE_JM]

Thank you thank you EulerFan101
you are the BEST 
And thanks to everyone who has ever helped me here. I always feel a bit hesitant to ask my teachers. I don't think I could have understood this much without all of you!

[keltingmeith]

I was looking at the chapter summary where it said that  b^(log_ba)x is equal to a^x
How is this so? Can you just bring it down to log_bax * b despite it is an exponential?
Thanks,

Erm... Not quite.

[SE_JM]

Erm... Not quite.

Aah... Is there a rule that if a^log_a=1?

if there is, I've never seen one in the textbook   

Thanks

[keltingmeith]

Aah... Is there a rule that if a^log_a=1?

if there is, I've never seen one in the textbook   

Thanks

Yep - for all f(x)>0

[lzxnl]

Yep - for all f(x)>0

With any extension of the natural log to complex numbers, I think that also works for any complex f not equal to zero

[keltingmeith]

With any extension of the natural log to complex numbers, I think that also works for any complex f not equal to zero

Alright, I'll let you know the next time I see a complex function in a methods paper.