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# Recent problems solved by 'longjonsilver'

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 Functions/30491: Let F(x)= x^3-2x+1, and g(x)=x^5-x+3. Find F^-1(x) and g^-1(x). Please help me thank's1 solutions Answer 17194 by longjonsilver(2297)   on 2006-03-17 06:26:06 (Show Source): You can put this solution on YOUR website!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.
 Linear-equations/30535: A bus has its maximum number of passengers when it leaves the bus station. At the first stop, half of the passengers get off. At the next stop 7 people get on and at the next stop 16 people get off. There are now 17 people on the bus. 1 solutions Answer 17193 by longjonsilver(2297)   on 2006-03-17 06:19:15 (Show Source): You can put this solution on YOUR website!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.
 Graphs/30502: sketch the graph of , not by plotting points but by starting with and using translations. Describe the steps you take in getting the graph of y. Thank you!1 solutions Answer 17183 by longjonsilver(2297)   on 2006-03-17 04:22:05 (Show Source): You can put this solution on YOUR website! 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.
 Linear-systems/30514: I need help with a linear function table. Here's the table: x f(x) -6 -15 1 -1 7 11 11 19 I tried to find the pattern for the y's and the x's so that I could find the slope, but it doesn't seem like there is a pattern. I need to write the rule for the linear equation. Can you help? Thanks1 solutions Answer 17182 by longjonsilver(2297)   on 2006-03-17 04:14:33 (Show Source): You can put this solution on YOUR website!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.
 Equations/30481: Can someone please explain how to answer this question? (1/3)(2h-5)=1/4(3h-15) The answer I keep coming up with is h=-7.08 but when I check it, it does not equal. Thank you!1 solutions Answer 17121 by longjonsilver(2297)   on 2006-03-16 13:24:10 (Show Source): You can put this solution on YOUR website! mutliply both sides by 12... --> h = 25 check this answer. jon.
 Human-and-algebraic-language/30471: In What postive base B does the equation 4*12=103 for multiplication of base B hold?1 solutions Answer 17113 by longjonsilver(2297)   on 2006-03-16 10:58:10 (Show Source): You can put this solution on YOUR website!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.
 Linear-systems/30467: Please help me work this problem. I have the answer but don't know how to work it. Please explain in very plain English terms! Point A (-4,1) is in the standard (x,y) coordinate plane. What must be the coordinates of Point B so that the line x = 2 is the perpendicular bisector of AB ? (note: there is a line over AB) Thank you for your help...Karen1 solutions Answer 17110 by longjonsilver(2297)   on 2006-03-16 10:43:01 (Show Source): You can put this solution on YOUR website!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.
 Mixture_Word_Problems/30391: Hello I need help with a solution problem. I can not seem to put this thing together. A pharmacist is to prepare 15 milliliters of special eye drops for a glaucoma patient. The eye-drop solution must have a 2% active ingredient, but the pharmacist only has 10% solution and 1% solution in stock. How much of each type of solution should be used to fill the prescription. Help with this will be greatly appreciated, I am lost on this one. Thanks Terry1 solutions Answer 17094 by longjonsilver(2297)   on 2006-03-16 06:48:11 (Show Source): You can put this solution on YOUR website!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.
 Numeric_Fractions/30385: (1/a + 1) / (1/a2 - 1) The book gives an answer as: a/1-a, but I just can't see how they get it. You must first multiply both numerator and denominator by the LCD, right? Isn't this 2a? When I multiply both by it, I get: (2 + 2a) / (1 - 2a) and get stuck there. What am I doing wrong?1 solutions Answer 17092 by longjonsilver(2297)   on 2006-03-16 06:18:46 (Show Source): You can put this solution on YOUR website!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.
 logarithm/30416: Could you please help me with this problem. Thank you. * = 2 1/21 solutions Answer 17088 by longjonsilver(2297)   on 2006-03-16 05:17:48 (Show Source): You can put this solution on YOUR website!i am not sure what you what to find or prove here. What i can find for you is the value of k jon.
 Miscellaneous_Word_Problems/30425: What is the number of different 7 digit numbers that can be made by rearranging the digits 3053354?1 solutions Answer 17087 by longjonsilver(2297)   on 2006-03-16 04:46:20 (Show Source): You can put this solution on YOUR website!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.
 Geometry_Word_Problems/30440: An isosceles triangle has a perimeter of 8 cm. Find a function that models its area A in terms of the length of its base b.1 solutions Answer 17086 by longjonsilver(2297)   on 2006-03-16 04:36:53 (Show Source): You can put this solution on YOUR website!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.
 Exponents/30449: I'm not sure if I am posting this in the right area. But I need to: For the indicated functions f and g, find the functions f+g,f-g,fg, and f/g and find their domains. f(x)=3x g(x)=x^2+4 Could you show me how this is done. I did it using word and cannot paste the answer in here and need something to check my work by. Thank you so much!1 solutions Answer 17085 by longjonsilver(2297)   on 2006-03-16 04:12:10 (Show Source): You can put this solution on YOUR website!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.
 Linear-equations/30433: is the graphing method of solving equations when u figure out the x and y intercept then the slope and graph?? if thats not how you do it then how do you do it??1 solutions Answer 17084 by longjonsilver(2297)   on 2006-03-16 04:03:23 (Show Source): You can put this solution on YOUR website!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.
 Equations/30450: How do I find the correct way to solve the following problem? It is estimated that the Earth is losing 4000 species of plants and animals every year. If S represents the number of species living last year, how many species are on Earth this year?1 solutions Answer 17083 by longjonsilver(2297)   on 2006-03-16 03:57:59 (Show Source): You can put this solution on YOUR website!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.
 Graphs/30344: Find the x-intercept of the graph of y=3+6x1 solutions Answer 17005 by longjonsilver(2297)   on 2006-03-15 09:22:36 (Show Source): You can put this solution on YOUR website!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.
 Linear_Algebra/30338: Solve: x^2+y^2=16 x^2/25 + y^2/16 =1 (a) no solutions (b) [(0, -1), (0,4)] (c) [(0, -5), (0,5)] (d) [(-5, 0), (5, )]1 solutions Answer 17004 by longjonsilver(2297)   on 2006-03-15 09:21:05 (Show Source): You can put this solution on YOUR website!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.
 Functions/30254: evaluate each function rule for x= -2 f[x]=x-2 f[x]= 1/2x+2 1 solutions Answer 16995 by longjonsilver(2297)   on 2006-03-15 04:14:52 (Show Source): You can put this solution on YOUR website! 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