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I have plotted both curves for you. I have done this to show you that picking certain y-values gives you multiple values of x. This contravenes the one basic requirement of any function...it must NOT be one-to-many.





Hence, strictly, neither of these functions have inverse functions. Now, you could get round that by narrowing the domain of the functions, such that the one-to-many region of the curves is not included, but forgeet that: just say:

There is no inverse because the function is ONE-TO-MANY. If your teacher argues, then pass them over to me here and i shall argue with them.

jon.

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Let x = number of passengers at start.

((x/2)+7)-16 = 17
(x/2)+7 = 33
x/2 = 26
x = 26*2
x = 52

jon.

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Starting with , first we move the curve by 1 unit to the left to give , as seen by

Next, we stretch it by a factor of 3, anchored still on the x-axis at (-1,0). Look at the graph here and note the y-intercept is now at y=3, instead of y=1:

Finally, we will move the whole curve up by 2 units, so that the vertex is now at (-1,2) and the y-intercept is at y=5:

jon.

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you say it is a linear function, so in that case, we are looking for a "times table".

Pick any 2 values of x... say 7 and 11. They differ by 4. Their F(x) values are 11 and 19, which differ by 8.

So, x increases by 4 whilst y increases by 8... looks like the 2 times table to me.

--> f(x) = 2x

Now, it isn't THE 2x table: 2,4,6,8,10,... etc, but it is this shifted.

So, when x=7, f(x)=2x gives f(x)=14. Our function actually gives 11, so we need to subtract 3: f(x)= 2x-3.

CHECK:
when x=-6
f(-6) = 2(-6) - 3
f(-6) = -12 - 3
f(-6) = -15... CORRECT

You can then tell what the slope of the line is from y=2x-3

jon.

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mutliply both sides by 12...







--> h = 25

check this answer.

jon.

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as is usually the case, those students who do not understand something in maths, invariably do much harder things than is actually needed.



First, to mutliply the second bracket by the -1 written before it... the full writing of the question should be:

but mathematicians are lazy and we miss out some bits. The thing is we all know they have been missed out. Trouble is, when you start maths, you do not know it has been missed out, especially if your teacher doesn't teach it that way.



Basically, all that has happened, is that all the signs have been flipped.

Now, we can collect like terms... we have and and and terms with just numbers.

So, how many terms are there? Answer is 1 -->
how many terms are there? Answer is 3+4 -->
how many terms are there? Answer is -7-5 -->
how many pure numbers? Answer is 8

So, the answer is .

And i hope you can see that this is a lot easier than what you were trying to do.

jon.

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the answer is base 5.

I am not sure if there is a "proper" way of figuring this out, using "modulo" maths, but here is my approach:

12
x4
__

In base 10, we would say... what is 2x4? answer is 8.

In our base, the answer is still "8", but we will write it differently.

Seeing as how we have the number 4, the base must be base 4 or greater, so i started there:

base 4, the number 8 would be 20...no end "3" there. So, base 5...

base 5, the number "8" would be 13.. voila, a "3". So base 5 looks a good guess.

so, 2x4 is "8"...written as 3 with the 1 carried.

Now, 1x4 is 4 plus the carried 1 is 5. In base 5, 5 is written as "10"

So we have "10"..."3" --> 103.

Hope this explains it well enough for you.

jon.

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this is a very simple question really, just hidden in a lot of jargon.

Imagine the vertical line at x=2.
Imagine the point (-4,1)

Sketch a little hand-drawn graph of these.

Now, for the line x=2 to be perpendicular (ie at right angles) to the line AB, AB must be horizontal...since x=2 is vertical.

As this horizontal line passes through (-4,1), the equation of the line AB is y=1. However, the question doesn't ask for that. It asks for the point on the line AB (equation y=1) so that x=2 is a bisector.

Bisector means it cuts the line AB into 2 equal parts.

So, point (-4,1) is how far from x=2?... x=-4 and x=2? the seperation is 6.

So, the point B, on the other side must also be a distance of 6 away from the x=2 line. This means it mus be ay x=8. So the point is (8,1)

jon.








jon.

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First off, real my Lesson on this website on how to do this type of question.

Second, just looking at the question, we have some 10% and some 1% that mix to make a 2% solution.

Just by common sense, we will need more of the 1% solution, since the final answer is a 2% solution. Hoepfully that makes sense? We expect a lot more of the 1% solution to be used.

Right...
Let x = amount of 1% solution used.
So, how much of 10% is used? Answer is (15-x) --> ie everything else to make up the final 15ml.

BEFORE: x of 1% AND (15-x) of 10%
--> 0.01x + 0.10(15-x)

AFTER: 15ml of 2%
--> 0.02*15

And being a mixture, the BEFORE has to equal the AFTER:
0.01x + 0.10(15-x) = 0.02*15
0.01x + 1.5 - 0.10x = 0.30
-0.09x = -1.2
x = -1.2/-0.09
x = 13.3333 ml

and so there are 13.3333ml of the 1% solution and 1.6666ml of the 10% solution.

jon.

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Assuming you mean: , we get:





As for the answer you quote, a/(a-1), I cannot see how that was gotten, from the starting point i used.

jon.

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i am not sure what you what to find or prove here.

What i can find for you is the value of k










jon.

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if all the numbers were different, we would have 7! permutations... which is 5040 different numbers.

Additional thing to think about would be numbers like 0123456.. this is not a recognised number, so we cannot have the zero in the "first position".

So...
First position: how many numbers could we have? Answer is 6 (as the zero is not allowed there).

Second position: how many numbers? Well any of the 6 remaining.
Third position: Any of the 5 remaining
fourth position: Any of the 4 remaining
and so on.

Total: 6*6*5*4*3*2*1 --> 6*6! --> 6*720 --> 4320.

Notice, this is 720 less than 5040 ie it is one whole block of 6!

Now, this is not the final answer, since there are three 3's and two 5's, so there is going to be repetition of certain numbers.

Total number of different numbers = 4320/(3!2!)
Total number of different numbers = 4320/(6*2)
Total number of different numbers = 4320/12
Total number of different numbers = 360

So, there are just 360 different numbers that can be made up from the given set of digits.

jon.

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Draw an isosceles triangle and drop a perpendicular from the apex to the mid-point of the base, b. You now have 2 right-angled triangles.

Let the height = h
Let both the angled sides of the isosceles triangle be x, so that the perimeter is x+x+b --> 2x+b=8 which means that x=(8-b)/2.

What we need to do is rewrite the x in terms of b, to satisfy the question.

We have 3 things we can use:
1. Pythagoras' Theorem for one of the right-angled triangles
2. Area of a triangle is (1/2)*base*height
3. the relationship

1. In either of the 2 rightangled triangles, we have .

This means that
-- eqn1

Now, from , we have . So . Therefore

Right, we now have an equation relating x to b. If we substitute this into eqn1, we get:





so that

Right, we now have height in terms of just b.

So, from Area of a triangle is (1/2)*base*height, we get



--> this is the useful line of maths to understand. From here, we get:







I think this is a correct function for the area of the isosceles triangle when the perimeter is 8cm.

Jon.

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As to the domains, you need to know the domains of the originals to base them upon. You haven't supplied this info, so i assume both f(x) and g(x) have domains of "any real value".

f+g -->
f+g --> and its domain is also "any real value of x" since we CAN put in any value we like and get a valid y-value out.

f-g -->
f-g -->
f-g --> and its domain is also "any real value of x" since we CAN put in any value we like and get a valid y-value out.

fg -->
fg -->
fg --> and its domain is also "any real value of x" since we CAN put in any value we like and get a valid y-value out.

By f/g, i assume you mean...
f/g --> in which case the domain is again "any real value of x". If you have done Complex Numbers, there are 2 complex numbers that are NOT allowed, since we are trying to avoid the denominator being zero, because when that happens, the fraction zooms off to infinity, which is a bad thing mathematically.

jon.

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To me, when a question say "graph" or "plot" it means get some graph paper and work out a few points from the equation and then place them on the graph paper and join them up.

It it said "sketch", then i would find certain key points, like the axes intercepts and then do a rough drawing on my normal paper armed with this knowledge and an understanding of the basic shape of the curve.

You decide which you think you are being asked.

Jon.

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well?

if it looses 4000 a year and there were say 20,000 last year. How many this year? Answer is 16,000. How did i do that? So apply that to "variables" rather than numbers.

You try.

Jon.

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the x-intercept... where the line crosses the x-axis. The x-axis is the line y=0, so put this piece of information in. We get:

y=3+6x
0=3+6x
6x = - 3
x = -3/6
x = -1/2

jon.

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i believe that the answer is b) except you have written the answer incorrectly.

and


. Sub this into the second equation, to give:






-->
so x = 0.

If x=0, then from , we get
-->
--> or

so, y = 4 or -4.

The 2 solutions are (0,4) and (0,-4).

Check in the second equation,

using (0,4):


which is indeed 1... correct.

using (0,-4):


which is also 1... correct

jon.

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Not sure what you meant to write for the second one...it is ambiguous. You need to use brackets to collect terms together to remove the ambiguity. I shall assume you meant:








If you meant to write then learn to use brackets more in your maths language and now you can do this one :-)... the answer is -1/2.

jon

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To plot...pick values of x... your choice of values. I shall pick, -3, -2, -1, 0, 1, 2, 3.

For each of these, put it into to find what y is.

You will end up with 7 sets of x and y values. Plot these.

The question is asking you, where does the curve have y=0.. ie where does it cross the x-axis. That is all.

So, read off the value(s) of x where the curve crosses. If it crosses between integer values of x, say somewhere between x=2 and x=3, then they want you to write:

The solution lies between x=2 and x=3.

This is because your graph will not be accurate enough to quote the precise answer, so you reading the value from your graph as x=2.3, for example, is a waste of time.

Oh yes, if you chose x-values and when you plot the graph you missed the point(s) that cross the x-axis, then tough... pick some more x-values and plot those too.

Basically this will teach you three things hopefully:
1. What an equation physically looks like
2. What "solving" physically means to an equation.
3. Graphs are a pain to plot and are not accurate anyway.

jon.

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if then c has 2 possible values, either or .

Either use your calculator to find the decimal version of this number, which you will have to round up. Hence it will not be the precise answer, merely an approximation --> 36.125 to 5 significant figures.

Alternatively, leave it in radical form, but simplify the number if you can. In this example, we can, since 9 goes into 1305.






So the EXACT answers are and

Jon.

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. I hope this is what you meant?

Taking logs on both sides, and using natural logs, since we have the number "e"...







So, this is as far as you need to go...the question asks for it in this form. We could go further and collect terms together, however.

jon.

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Well, first lets do 72/8 --> 9


Now for the x terms. We have powers and we are dividing, so by the rules of exponents/powers/indices, the powers will subtract: is which is .

--> which MEANS . This is the final answer, written as a radical (or surd).

jon.

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i am not sure waht you mean...you have written something that is ambiguous.

You could mean in which case we have:

or . Now divide by -1
and hence

or you meant in which case we have:
. Now divide by -1
and hence

Both situations use the same rules, namely if you multiply or divide by a negative number, the inequality sign swaps round.

jon

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Look at the Lesson i have written on Straight lines... or Linear equations. It will help you to understand.

Jon.

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4 - (6 + x) = 13
4 - 6 - x = 13
-2 - x = 13
-x = 15
x = -15

jon.

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Let x = amount of pure acid added.

So how much of the 10% acid was there? Easy... everything else to make up the 54litres...so (54-x).

Now we say BEFORE = AFTER

BEFORE is (1.00*x) + (0.10*(54-x))
AFTER is 0.20*54

so (1.00*x) + (0.10*(54-x)) = 0.20*54
x + 5.4 - 0.1x = 10.8
0.9x + 5.4 = 10.8
0.9x = 5.4
x = 5.4/0.9
x = 6 litres

So we need 6 litres of the pure acid and also 48 litres of the 10% acid solution.

jon.

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you try this one... follow the process i did on the previous example.

jon

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you do not define your variables, so i am not 100% sure what v or c or R are. To me, c is the speed of light, so i need to know what you think it is here.

Jon.

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x + y = 5 --> y = -x + 5
x - y = 1 --> y = x - 1

Written in this form, the coefficient of x is the gradient... we have -1 and +1. They are not parallel.

If (grad1)*(grad2) = -1 then the lines are perpendicular.

-1*1 is -1... so the 2 lines are perpendicular.

jon.

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line (-1/3)x-y=2 --> y = (-1/3)x-2, so its gradient is (-1/3). Any line perpendicular to this has gradient of 3 (2 perpendicular gradients, m*n = -1)

So we have x=-5, y=-1, m=3...find c from y=mx+c and that will give you the equation of the line.

jon.