When finding the domain of a function, remember it's all values for the variable where the function is defined.

For example, f(x) = 1/(x - 2), the domain is all real nubmers except for 2. A value of 2 for x makes the denominator 0 and the function undefined.

Be careful with some functions. Make sure everything is simplified before determining the domain. If

f(x) = (x^2 - 16)/(x - 4) it might be tempting to see the denominator and conclue the domain is all real numbers except for 4. But this is incorrect. 

If we factor the numerator, we get (x -4)(x + 4).  The (x - 4) in the numerator will cancel with the (x - 4) in the denominator.

Therefore f(x) = x + 4 and the domain is all real numbers.

When working with formulas involving Pi, sometimes it's easier to just leave the answer in terms of Pi. Othertimes, when needing a specific value for volume or area of a sphere, circle, cylinder for example, it's best to use 3.14 or 22/7 as approximations for Pi. Yes, many people have memorized Pi to a ridiculous number of digits, but for all practical purposes, a few decimal points of Pi will suffice.

Check out the math book as an ebook in pdf form to be viewed on the computer and also as a paperback. 

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When using matrices to solve a system of three equations in 3 variables x, y and z, write the system as an augmented matrix and use matrix row operations to get the matrix into row-echelon form. We use what is called Gaussian elimination.

When the matrix has 1's along the diagonal from upper left to lower row and 0's underneath the 1's, we can use the value obtained for z and substitute in for z in the second equation to solve for y and then substitute y and z in the first equation to solve for x.

Systems of equations may have none, one or infinitely many solutions. 

When solving for square roots and cube roots, it's very helpful to know many of the perfect squares and perfect cubes.

The first 25 perfect squares are

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625

The first 10 perfect cubes are

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Knowing these will be very helpful and they aren't hard to memorize.

To find the inverse of a function, substitute y for f(x), then interchange x and y and solve for y.

For example

If f(x) = x + 7, replace f(x) with y to get

y = x + 7

Next interchange x and y to get

x = y + 7

Solving for y we get

x - 7 = y

The inverse is x - 7

Recall that we worked with systems of linear equations such as the following:

3x + 4y = -6

5x – 2y = 1

5x + 6y – z = 10

7y + 4z = 3
z = -1

We can use a matrix to rewrite these systems in a simpler way. We use what is called an augmented matrix, which has a vertical bar separating the columns of the matrix. The group on the left of the bar are the coefficients of each variable and the group to the right of the bar are the constants (the numbers after the = in each equation in the system). In the systems above, the augmented matrices are as follows:

To solve a system of linear equations in three variables, we wish to produce a matrix with 1's along the diagnoal from the upper left to the lower right of the matrix, with 0's undereath the 1's. Such a matrix will look as follows:

The letters a through f represent real numbers. Recall that the elements of the augemented matrix to the left of the vertical bar represent the coefficients of the variables. Therefore in the above augemented matrix, we can conclude that

x + ay + bz = c

y + dz = e
z = f

Since we know the value of z, we can subsitute that into y + dz = e to solve for y. Then we can substitute the values for y and z into x + ay + bz = c and solve for x.

When trying to remember the ratio for each trigonometric function,  sine, cosine, tangent, cotangent, secant and cosecant, we really only need to know sine, cosine and tangent. Once we know them, cotangent, secant and cosecant are the recipricol of tangent, cosine and sine, respectively.

In a right triangle, the sine of an angle is the length side opposite the angle divided by the length of the hypotenuse (the side opposite the right angle).

Cosine is the length of the side adjacent to the angle divided by the length of the hypotenuse.

Tangent is the length of the side opposite divided by the length of the side adjacent.

Sine = opposite/hypotenuse

Cosine  = adjacent/hypotenuse

Tangent = opposite/adjacent

Cotangent = 1/tangent

Secant = 1/cosine

Cosecant = 1/sine

Notice that tangent is sine/cosine

To answer yesterday's problem, have to know that tan b = sin b/ cos b

Therefore,


sin b/tan b = sin b/ (sin b/cos b)

                  =  sin b/1  * cos b/sin b

                   = cos b

This will be easy for those of you that know trigonometry.

When do we add exponents and when so we multiply exponents?

If we have a variable raised to a power times another variable raised to a power, we add the exponents.

For example,  x^2 * x^5 = x^7

If we have a variable raised to a power and that quantity is raised to another power, then we multiply exponents.

For example,  (x^2)^5 = x^10

It's easy to illustrate this.

x^2 * x^5

Think of x^2 as x * x and x^5 as x * x * x * x * x

Notice there are 7 x's total, therefore when multiplying it's x^7.

(x^2)^5

Think of x^2 * x^2 * x^2 * x^2 * x^2

and x^2 = x * x

Therefore we now have x * x * x * x * x * x * x * x * x * x = x^10

Remember to check out my book at lulu.com. The book is a self help Algebra book covering 1st and some 2nd year high school school algebra concepts. It's also good for a beginner college algebra course. The book is available as a pdf for pc viewing and in paperback.

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 I'll show you 2 different methods to multiply 2 numbers.

   23
x 15
 115    (multiply 5 by 3 to get 15, carry the one, 5 times 2 is 10 and 1 is 11)
 230   (add a zero under the 5 and multiply 1 by 3 to get 3 and 1 by 2 to get 2)
 345   (add the numbers)

This method is probably the one that the majority of people learned in school.

Here's an alternate method.  We will think of 23 as 20 and 3 and 15 as 10 and 5

Multiply 20 by 10 to get 200
Multiply 20 by 5 to get 100
Multiply 10 by 3 to get 30
Multiply 5 by 3 to get 15

Now we add the numbers to get 345.

Essentially we are using the distributive property twice.

 20(10 + 5) + 3(10+ 5)

200 + 100 + 30 + 15 = 345

What's interesting is we can also distribute as follows and get the same answer

10(20 + 3) + 5(20 + 3)

200 + 30 + 100 + 15 = 345.

Remember when adding or subtracting matrices, the matrices must be the same size.  Corresponding elements are added or subtracted to give elements of the new matrix.

Example: 
         1  3  5
A =   5  7  8
         2  0  9

         -2  8  0
B =    3  1   1
          5  0   2

                1-2   3+8  5+0       -1  11  5
A + B  =  5+3  7+1  8+1  =    8    8  9
                2+5  0+0  9+2        7    0  11

For those having difficulty calculating the amount of tip to leave at a restaurant, here's a quick and easy way. If you want to leave 15%, take 10% and then half of that 10% and add them together.

Total bill : $24

10% = 2.40
+ 5%   1.20
Tip =   $3.60

If you wish to leave 20% and you don't feel like multiplying, just take 10% and double it.

10% is simple to get. Just move the decimal point over 1 to the left. 10% of 25.78 is 2.58 (rounded off).

When multiplying matrices, remember the number of rows in the first matrix must equal the number of columns in the second matrix.  See how this multiplication occurs in the example below.

If the number of rows in the first matrix does not equal the number of columns in the second matrix, then the matrices cannot be multiplied. See the example below.

Matrix A has 2 rows and matrix B has 3 columns. Therefore these matrices cannot be multiplied together.

Here's a tip for working with formulas involving summations. I was working with a student yesterday who was getting confused with squaring a sum of values and squaring each individual value and then summing. This caused numerous errors when calculating the value of the correlation coefficient.

If the formula asks for sum x^2 then you square each x value and then take the sum. 

For example, if the x values are 2, 3, 5, 6, 9 square each value first

2^2 = 4
3^2 = 9
5^2 = 25
6^2 = 36
9^2 = 81

Now get the sum of these values

4 + 9 + 25 + 36 + 81 = 155


If the formula asks for (sum x)^2, you get the sum of the x values, then square the result.  In the above data set, we get

2 + 3 + 5 + 6 + 9 = 25, then square 25 to get 625.

The key to remember is to perfom everything inside the parentheses first.  Recall the order of operations is

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

You can remember this by remembering the phrase "Please Excuse My Dear Aunt Sally"

Front and back cover artwork for book.

Continuing on the theme of probability, suppose you have 12 songs on a CD and you want to play them in random order. How many ways can those songs be played and what is the probability that the song order on the CD will be the order played?

The song order on the CD is only 1 possible way the songs can be played. The probability is then 1 divided by the number of possible ways all songs could be played.

Have 12 slots, each representing a song selection  ___  ____ ____ ___ ___ ___ ___ ___ ___ ___ ___ ___

In the first spot, any of the 12 songs could be played, then in the second spot any of the remaining 11 songs could be played, in the next spot any of the remaining 10 songs could be played and so on.  Therefore the possible ways the songs could be played is 12 * 11 * 10 * 9 ..... * 1.  This is represented as 12!.  The probability is then 1/12!

Suppose you flip a fair coin (probability of heads = probability of tails) 6 times and the coin lands "tails" face up each time.  What is the probability that the next toss will also be a tail? What is the probability that the next toss is a head?

One might think that it's more likely to get a head since the first 6 tosses were tails, but in fact the probability that the next toss is a tail is .5, as is the probability that the next toss is a head.  The coin has no memory, so to speak. Even if the first 100 tosses landed tails up, the next toss still has a 50 percent chance of landing tails up.

Now the probability that all 6 coins tossed landing tails up is (1/2)(1/2)(1/2)(1/2)(1/2)(1/2) = 1/64.

Each toss of the coin is an independent event. What occurred previously on the coin toss has no bearing on the outcome of the next coin toss.