Inequilities, Matrices, Functions

• Lord dejavu

  27 Sep 09, 08:15

  :(
• Lord dejavu

  27 Sep 09, 08:19

  Hi all. Some questions i cant figure out on the above mentioned topics. Inequilities: solve for x(x+4)(x-1)>4x(x+2). hence, deduce the solution to x(x+8)(x-2)>8x(x+4) I'm able to do the first part. But for the second part, how do i know what to replace x by? Is there any "golden rule" to follow when it comes to dealing with this kind of questions? Matrices: I think some of the word problems can be a bit tricky, as in i dont know how to form the correct matrix. Take for example: A food manufacturer invents a new flavour of meatball from several ingredients. The main ingredients are beef, chicken and fish. 100g of beef contributes 20g of proteins, 30g of fats and 0.5g of cholesterol. 100g of chicken contributes 45g of proteins, 30g of fats and 0.2g of cholesterol. 100g of fish contributes 75g of proteins, 20g of fats and 0.2g of cholesterol. The key nutritional requirements for each 100g of mixture of meatball are 60g of proteins, 20g of fats and 0.2g of cholesterol. Other ingredients do not contribute to these three key nutritional requirements. Find the amount of beef, chicken and fish the food manufacturer should include in 1 kg of mixture to meet the nutritional requirements. First of all, i let the amount of beef, chicken and fish be x,y, and z respectively. Then after that i'm stuck. Functions: f(x) |---> ln(x+2) 0. Find f^-1 in similar form. ans: (e^x)-2 ln2
• wee_ws

  27 Sep 09, 09:11

  Originally posted by Lord dejavu:

  Hi all. Some questions i cant figure out on the above mentioned topics.

  Inequilities: solve for x(x+4)(x-1)>4x(x+2). hence, deduce the solution to x(x+(x-2)>8x(x+4)

  I’m able to do the first part. But for the second part, how do i know what to replace x by? Is there any “golden rule” to follow when it comes to dealing with this kind of questions?

  Hi,

    Some common substitutions are:

    |x|, e^x, ln x, 1/x, -x, x + a (a is some constant), etc.

    Thanks!

  Cheers,
  Wen Shih

• wee_ws

  27 Sep 09, 09:18

  Originally posted by Lord dejavu:

  Hi all. Some questions i cant figure out on the above mentioned topics.

  Functions:

  f(x) |-> ln(x+2) 0. Find f^{-1 in similar form. ans: (e}x)-2 ln2

  I am taught that the domain of f^-1 is the range of f(x). My problem is finding the range of f(x). My gc (TI-84 Plus) doesnt give me the correct range for some reason. From the graph in the gc, i noticed that the range in is between -3 to 3, but how do i get the exact value of ln 2 and ln 7? Also, is there another way of finding the range of a function, especially those that deal with logarithms without the use of the gc? I realize it has its limitations when dealing with functions questions involving log, expo and trigo.

  Thanks, really appreciated.

  Hi,

    Never use GC for it has limitations in displaying graphs, like what you have already encountered.

    The graph of y = ln (x + 2) can be sketched from the basic graph of y = ln x (covered in O-level and should therefore be memorised), which is translated 2 units to the left. Based on the range of values of x given by the domain of f, one may use the graph to obtain the range of values of y, by direct substitution.

    Thanks!

  Cheers,
  Wen Shih

• wee_ws

  27 Sep 09, 09:38

  Originally posted by Lord dejavu:

  Hi all. Some questions i cant figure out on the above mentioned topics.

  Matrices: I think some of the word problems can be a bit tricky, as in i dont know how to form the correct matrix. Take for example:

  A food manufacturer invents a new flavour of meatball from several ingredients. The main ingredients are beef, chicken and fish.

  100g of beef contributes 20g of proteins, 30g of fats and 0.5g of cholesterol. 100g of chicken contributes 45g of proteins, 30g of fats and 0.2g of cholesterol. 100g of fish contributes 75g of proteins, 20g of fats and 0.2g of cholesterol. The key nutritional requirements for each 100g of mixture of meatball are 60g of proteins, 20g of fats and 0.2g of cholesterol. Other ingredients do not contribute to these three key nutritional requirements.

  Find the amount of beef, chicken and fish the food manufacturer should include in 1 kg of mixture to meet the nutritional requirements.

  First of all, i let the amount of beef, chicken and fish be x,y, and z respectively. Then after that i’m stuck.

  Thanks, really appreciated.

  Hi,

    Since we have 3 unknowns to be found, we will need to form 3 equations. Also, one needs to be clear about the measurement unit of x, y and z. In this case, we use gram as the unit.

    Now read the question carefully to see possible relationships which may be formed.

    We need 1 kg (1000 g) of mixture. For this amount of mixture, how much proteins, fats and cholestrol will there be?

    For proteins in 1000g of mixture, how much is contributed by each of x, y and z? That will give us an equation.

    For fats and cholestrol in 1000g of mixture, ask the same question. That should give us another 2 equations.

    This process of question analysis can be applied to other questions of a similar nature.

    Thanks!

  Cheers,
  Wen Shih

• Lord dejavu

  27 Sep 09, 22:39

  Originally posted by wee_ws:

  Hi,

    Some common substitutions are:

    |x|, e^x, ln x, 1/x, -x, x + a (a is some constant), etc.

    Thanks!

  Cheers,
  Wen Shih

  
  Thanks Mr Wee, do you have the solution for the inequility question? The expression with the smiley is suppose to be x(x + 8)(x-2)>8x(x+4). I have tried the some of your suggestions but i cant seem to get the answer :(
• wee_ws

  28 Sep 09, 08:48

  Originally posted by Lord dejavu:

  
  Thanks Mr Wee, do you have the solution for the inequility question? The expression with the smiley is suppose to be x(x + 8)(x-2)>8x(x+4). I have tried the some of your suggestions but i cant seem to get the answer :(

  Hi,

    The original inequality is x(x + 4)(x - 1) > 4x(x + 2).

    The new one is x(x + 8)(x - 2) > 8x(x + 4).

    We start by looking at the sides individually.

    LHS: Compare x(x + 2) with x(x + 4). It seems that 2 has been doubled to 4.

    RHS: Compare x(x + 4)(x - 1) with x(x + 8)(x - 2). It seems that 4 and -1 have been doubled to 8 and -2 respectively.

    We now guess that the substitution is x/2.

    Let's verify our guess:

    x(x + 4)(x - 1) > 4x(x + 2)

    => x/2 (x/2 + 4)(x/2 - 1) > 4 (x/2)(x/2 + 2), upon substitution

    => x/8 (x + 8)(x - 2) > x(x + 4)

    => x(x + 8)(x - 2) > 8x(x + 4), as desired.

    Let's add another item to the list of substitutions above: x/a, where a is some constant.

    The above reasoning process is always used to guess the substitution, when the standard ones do not work. Do internalise this strategy.

    Thanks!

  Cheers,
  Wen Shih

• Lord dejavu

  28 Sep 09, 22:27

  Mr Wee, how come you guess that the substitution is x/2 and not 2x? Since the constants of the old equation is doubled, shouldnt it be 2x? This part is kind of confusing. Thanks.
• wee_ws

  28 Sep 09, 23:55

  Hi,

    I use the fact that when I factor out 1/2, everything inside doubles. Thanks!

  Cheers,
  Wen Shih