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The domain is the set of all valid value for the 'independent' variable. The range is the set of the resulting 'dependent' variable.
So, let's say you had a function like
Then the domain is the set of all valid values for x and the range is the values you would get for all the 'y' when you subbed in all the 'x's.
As far a a function goes, let's say you want to go to a local theme park. Tickets are $25 and parking is $10.
The total price for a family is given by where a is the number of people in the family.
For that function the domain is all values of 'a' that are positive (no family has 0 or negative number of members).
The range is (35, 60, 85, 110, etc... like that)

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You have more than a clue. Don't fret, You are part way there already.
Your solution uses the number n. But there is nothing that ensures that 'n' is even. So let's use a different set of numbers.
Let's assume n is even, if so, then n/2 is still an integer.
Our three numbers will be n/2, (n+4)/2 and (n+8)/2
Add in one more given " four times the middle integer is equal to two less than the sum of the first and third integers."
so
Left left side is four times the middle number. The right side is the (sum of the first and third) minus 2
Now solve





n = -6
Check : is 4(-4) = (-6 + -2) - 2?
Nope, so I think there is not a solution to this one

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What does radius mean? The distance from the center of a circle to every point on the circle.
What does diameter mean? A line segment that connects two points on a circle and passes through the center of the circle.
So, the end points of a diameter are both on the circle and they are on the opposite each other on the circle. (directly across the circle).
Thus
d = 2r

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Beth, i don' claim to be a guru at logs, but here is what I get. Compare that to your book and please let me know what it says .





Does that agree with your book?

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There a several ways to do this one. The form "" a = bc + d where b and d are constant" is just a line (the familiar y = mx + b). So you could figure out a slope and then do some math to get to the intercept.
But let's look at this as a logic problem and find some patterns.
Every increase of 'c' by 1, results in an increase of 'a' by 3. You can see that by finding the difference between each successive value in the series of 'c' and 'a' numbers.
So that means .
Now instead of moving up the 'a' and 'c' series, move back one. Decrease 'c' by 1 (1-1) and get 0. Decrease 'a' by 3 (8-3) and get 5. So when c = 0, a = 5.
=
Which means
Now plug in b and d

What is 'a' when 'c' = 19? I leave that to you :-D

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The 'rules' are the same as solving equalities with a couple extras.
1)The main difference is that whenever you multiply or divide by a negative number, then the sign of the equality 'flips'. That is, a 'less than' becomes a 'greater than' and vice-versa.
2)Another thing that you need to look out for is:
If the inequality is 'absolute' (that is, greater or less, not greater/equal to or less/equal to), then when you plot your answers, make sure to use a dashed line instead of a solid one. Then shade the portion of the graph that is your solution set.

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2 cat, 2 mice, 2 days yields one mouse/cat/per day
If you have 20 cats, and 10 days, then you'll get = mice in that period.
That's a lot of mice. yikes!

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Let d be the number of dimes.
Let n be the number of nickels.
Given:
total number of coins = 218 = number of dimes + number of nickels =>

total value of coins = 16.05 = 0.10d + 0.05n (multiply the value of each coin by the number and then add the totla for both types of coins) -->
lets convert from dollars to pennies to make things easier (we just multiplied the eqaution above by 10)
You now have 2 equations and 2 unknowns, so you can solve this.







You have 115 nickles.
218-115 = 103 dimes
Check your answer
10.30 in dimes + 5.75 in nickels = $16.05. check!

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You're on the right track. But you need to establish the relationship between distance, time and speed.

The distance the whale swims is the same going both directions.
The rate changes based on swimming with or against the current
Let the calmspeed = x and the current =y
RateWith = calmSpeed+Current = x+y
RateAgainst = calmSpeed-Current = x-y

So
with current
against current
So


remember this for later



use the value for x we found above to get


x = 30+5 = 35

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Given :




Check your answer. Does 3^2.0964 = 10???

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Given:

So the LCD will be

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Great question.
Some folks are more 'visual' than others. Folks that need to see things do well with graphical representations. Also, students with a bent toward geometry like graph. Graph are also a great way to link algebra with geometry.

The quadratic equation will 'always get you an answer if there is one'. But it requires more arithmetic and, IMHO, presupposes that the user understands complex numbers -- since the question of 'what happens when the determinant is negative' is almost certain to be raised. On the other hand, the quadratic equation is easily programmed, so it is a great tool for computer solutions (and also more sophisticated calculators).

Completing the square is an 'OK tool'. But fact is, once one uses it to prove the quadratic equation, I have found few places (outside of problems that specifically state one must use completing the square) where I actually used that method.

Factoring is handy when the the equation are 'fairly simple'. Factoring is very important when polynomial division/multiplication is part of the problem. Plus, factoring is fun and helps students connect algebra to the underlying properties that also apply to arithmetic and also, later on, to more sophisticated algebras.
If you get other responses, can you do me the favor of compiling them and sending the merged ideas to me? I am interested in what others might think.

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Given : (4.2*10^-2) (3.5*10^8)
=
Think of it this way. = 1
10^1 = 10
10^2 = 100
10^-1 = 0.1
10^-2 = 0.01
You can see that each number has the decimal point moved. For positive numbers, the decimal point moves one place to the right with each increase of 1. With negative numbers, the decimal moves one place to left with each decrease of 1.
For scientific notation, the "number part" is always between 1 and 10.
So
is already in scientific notation. So is

Use the same technique to solve the second one. Ping me if you can't get it
The value of is 0.042 (decimal moved two place to the left.
The value for is 350,000,000 (deicmal 8 place to the right)
To multiply the numbers, use the scientific notation form.



You are almost there now. The only thing left to do is make the 'number part' be between 1 and 10.



Use the same technique for the second one. Just remember that when you divide, that is the same as subtracting exponents. Ping me by email if you can't get it.

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2.54cm = 1 inch
=
To convert 350 inch^3 into cm^3

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Let's get the answer in feet


Thats a long way. So don't EVER drink and drive or ride in a car with someone who has been drinking.
Also notice how the "units" in the problem canceled out. Many times, units (feet, hour, miles, etc) can help you validate if an answer is correct or not. If the units endup being something that doesn't "make sense", then it is likely you did something wrong along the way.

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I get 37.08 - so you are correct

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You are given two points. The distance between two points is given as:





distance = 5a

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Parallel lines have the same slope.
The slope of AB = =
The slope of CD = =

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Use the Pythagorean theorem

in this case, c=10 (the radius), a = 6 so solve for b
You can see this is a 3:4:5 triangle. So b = 8
The chord will be 8+8 = 16

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Use the quadratic formula.

a =1, b = -2, c =-6

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Given :


Add these two

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Given :



Use the quadratic equation

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This applies often to problems in physics -- and in real life.
So let's say an expression is given as the model for something based in time. Example, the trajectory of a batted ball.
If the question asked "when will the ball hit the ground?" and one of the answer is -5 seconds. An answer that happens 'before the ball is hit', is a solution that is unacceptable.

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Use the "vertical line" rule to help decide if an expression is a function. If a vertical line intersects the graph of an expression in more than one point, then that expression is NOT a function.
Now to your questions ;
Are all linear equation functions? No.
Is there an instance when a linear equation is not a function? Yes, a vertical line is linear, but fails the vertical line test.
Can you give an example of and nonlinear equation? A parabola is an example of a non linear equation y = x^2

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The sum of interior angles of a polynomial is given by

See http://www.geom.uiuc.edu/~dwiggins/conj07.html for a nice description and an example using 6 sides polynomial

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This is a line where every point on the line has a y coordinate of 8.
x can be anything, but y is always 8.
So it is a horizontal line that goes through point (0,8)

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Given :

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For problems like these, it is often helpful to scale small and look for patterns
I suggest you work the problem with only 10 lockers and 10 students, and then with 20 and 20.
Pay attention to the pattern you see. If you still can't figure it out, send me an email and I'll give you a hint.

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Given:

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given :


On this type of problem, since the coefficient for the a^2 term is 1, you are looking for factors of 3 that sum to 2. The factors of 3 are 1 and 3. 3-1 equal 2.
Make sense?