Higher Mathematics

zetafunc. · 2011-09-16 07:25:38

Remember, all I have done is replaced any 'x's with whatever g(x) was. So if g(x) was equal to 'stuff' then I would have

f(g(x)) = 4(stuff) + 2

bobbym · 2011-09-16 07:27:02

Hi lightning;

Wherever you see an x in f(x) you put in an x - 5

http://www.mathsisfun.com/sets/function … ition.html

lightning · 2011-09-16 07:51:24

Not sure how it's 4x - 18. Like, I can't see the jump? You know what I mean?

Last edited by lightning (2011-09-16 07:53:13)

lightning · 2011-09-16 07:54:17

Oh, nevermind. Thanks.

So what happens when It's like h(x) = f(g(x)) ?
I saw something like that. What's the difference?

bobbym · 2011-09-16 08:00:05

What is the difference between calling something x or y?

Bob · 2011-09-16 08:10:46

hi lightning

Have you sorted the trig question because I can give you exactly the diagrams you need.?

post back.

Bob

Last edited by Bob (2011-09-16 08:11:17)

lightning · 2011-09-16 08:17:44

bobbym wrote:

What is the difference between calling something x or y?

lightning wrote:

So what happens when it's like h(x) = f(g(x)) ?
I saw something like that. What's the difference?

Nothing. But there was a third one involved too so that one confused me in class.

bob bundy wrote:

Have you sorted the trig question because I can give you exactly the diagrams you need.?

I'll take all the help I can get. Fire away.

bobbym · 2011-09-16 08:24:00

h(x), the name of the function is h. f(x), the name of the function is f. Just a name. The most common is y = f(x), meaning y is a function of x, y depends on x. If you have ever graphed any equations by hand you know that you choose an x and that determines the y.

lightning · 2011-09-16 08:34:15

Ahh! So the answer would be;

h(x) = f(g(x))
h(x) = 4x - 18

I see, I thought it would be a little different with a third letter involved.

bobbym · 2011-09-16 08:39:27

Yep, a whole new function to play with called h. Remember functions are fun and I can prove it:

function.

Bob · 2011-09-16 08:51:18

hi lightning,

If you draw a random (any size for the sides) right angled triangle and measure the Opposite, Adjacent and Hypotenuse, then you can calculate the SIN, COS and TAN by the usual formulas:

etc

But you won't get very accurate values by measurement alone.

There are ways of getting values much more accurately and that's what a calculator will do.

Choose an angle , calculate it's SIN and you'll get lots of decimals places.

But there are a few trig values which come out exactly and that's what you have in post #3.

Have a look at my first diagram below.

I've drawn a 45, 45, 90 triangle.

As it is isosceles AB = BC. So when you do

That's 1 exactly, not 1.00000000001 OR 0.9999999999999.

So TAN(45) = 1

Now look at the second triangle.

It's an equilateral triangle, so 60, 60, 60. No right angle yet.

But split it down the middle and you get two identical 60, 30, 90 triangles.

Let's say the sides DE, EF and FD are all 2.

Then DG = 1

So we can work out SIN(30) as follows:

Also, what was opposite to 30, is adjacent to 60, so the same calculation gives

So there's another exact value. Not 0.4999999999 OR 0.50000000001, but 0.5 exactly.

I was always hopeless at remembering these, so my exam papers were always full of little scribbled diagrams with me working them out from scratch each time.

I seem to have typed a lot, so I'll pause there to let you digest that lot, and then go on to the ones with a square root sign in them.

Bob

Last edited by Bob (2011-09-16 08:54:41)

lightning · 2011-09-16 09:04:19

Thanks, that really helped. I'll practice using that equation over the weekend.

I don't think we've done roots. What do they look like? sin^2 = ? or something?

Bob · 2011-09-16 09:10:34

hi lightning,

No, I meant things like

The square root of a number has the property that if you times it by itself you get back to the number.

eg.

and

But try doing 1.732051 x 1.732051 and you won't get exactly 3, although you should get an answer that's pretty close.

If you get a calculator and key in for the square root of 3 you'll get

1.7320508
or
1.7320050808
or
...........

depends on how many digits the calculator can show.

So can we ever get the square root of three exactly?

Well not by writing out a long string of decimal digits. Mathematicians have shown that the digits of √3 go on for ever without ever stopping or recurring.

But mathematicians are happy to call √3 the exact value. When we write that we mean the number that when squared gives 3, even if we cannot write all its digits.

When you next post back I'll show you an example that will show the advantage of exact values.

Bob

Last edited by Bob (2011-09-16 09:24:12)

lightning · 2011-09-16 09:44:05

Ahh, I see. Post away.

Bob · 2011-09-16 10:16:55

hi again,

OK, I've made up a question using Pythagoras' theorem.

Firstly, I've got a triangle ABC with a right angle at B. (see diagram at end)

AB = 3 and BC = 2.

So we can work out AC.

Then use AC as a shorter side in a second right angled triangle ACD with the right angle at C.

CD = 6.

Using the value worked out for AC and the value for CD, work out AD.

Inaccurate answer.

Now for the exact answer.

Leave the answer like this.

Leaving the first answer as √13, pretending we know this number exactly, means that when we square it, we get back to 13. This is totally accurate; there's no rounding off error. So the final answer of 7 is exactly right.

Post back to let me know if you are still following me ok.

Bob

Last edited by Bob (2011-09-16 10:19:51)

lightning · 2011-09-17 06:37:36

I'm sorry, but you've lost me. We've not done this type of thing in school yet.
I would of rounded and got 7 anyway. But the whole double root thing is just over my head. I don't think we need to know that at moment. I dunno lol. But at least I know how to read the exact value triangles now. :3

Bob · 2011-09-17 09:52:46

hi lightning,

OK. I'll just finish off the exact trig angles stuff then. I've repeated the diagram below.

In the triangle ABC we can use Pythagoras' theorem to calculate the exact value of AC.

So

Then in the triangle DGF

so

Thus

and

Hope that's all clear.

Bob

Last edited by Bob (2011-09-17 10:02:47)

lightning · 2011-09-18 07:01:55

Hold on, why are you using sin and tan?

Bob · 2011-09-18 08:08:58

hi lightning,

The angles between 0 and 90 for which exact answers can be found * are

30, 45, and 60.

For these SIN, COS and TAN can be written. That's 9 values in total.

I did the missing ones in an earlier post.

* Actually there are more but they are not often used in this sort of course.

Bob

lightning · 2011-09-20 06:28:21

Oh I see. Thank you.

Math Is Fun Forum

#26 2011-09-16 07:25:38

Re: Higher Mathematics

#27 2011-09-16 07:27:02

Re: Higher Mathematics

#28 2011-09-16 07:51:24

Re: Higher Mathematics

#29 2011-09-16 07:54:17

Re: Higher Mathematics

#30 2011-09-16 08:00:05

Re: Higher Mathematics

#31 2011-09-16 08:10:46

Re: Higher Mathematics

#32 2011-09-16 08:17:44

Re: Higher Mathematics

#33 2011-09-16 08:24:00

Re: Higher Mathematics

#34 2011-09-16 08:34:15

Re: Higher Mathematics

#35 2011-09-16 08:39:27

Re: Higher Mathematics

#36 2011-09-16 08:51:18

Re: Higher Mathematics

#37 2011-09-16 09:04:19

Re: Higher Mathematics

#38 2011-09-16 09:10:34

Re: Higher Mathematics

#39 2011-09-16 09:44:05

Re: Higher Mathematics

#40 2011-09-16 10:16:55

Re: Higher Mathematics

#41 2011-09-17 06:37:36

Re: Higher Mathematics

#42 2011-09-17 09:52:46

Re: Higher Mathematics

#43 2011-09-18 07:01:55

Re: Higher Mathematics

#44 2011-09-18 08:08:58

Re: Higher Mathematics

#45 2011-09-20 06:28:21

Re: Higher Mathematics

Board footer