Functions and the aMAZEing Math Maze

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Ah, if you're here - I assume - you're having difficulties with functions.  They are, admittedly, a bit weird.  I will get to you're follow up question in a moment, but first let's talk about functions a bit.  They are really handy, and you'll be seeing a lot of them.

Why do we need functions?

We are replacing the y in a formula with f(x).  So, instead of y = 2x + 7, we'd like to start using f(x) = 2x + 7.  We are doing this because there's a lot of interesting things we can be doing with formulas, and some of them are less confusing if we used this notation.

For instance - we used to have problems like ...  Solve y = 2x + 7 when x = -3.  So, we'd substitute -3 for x and the equation becomes y = 2(-3) + 7 or y = 1.

But now, we would say solve f(-3) = 2x + 7  Looks different but it's the same problem f(-3) = 1.  It's even more accurate because y actually doesn't = 1.  It only equals 1 when x = -3.

O'k, just to make things a bit more complex.  We also have terms like g(x), h(x) etc.  We can have any letter where the f is.  We use this because we sometimes have more than one function.  f(x) = 2x + 7 and g(x) = 5x - 2.  would be an example.

If this still makes no sense, there's more explanation at Math Is Fun.

Why did I miss the original question?

Sure - right to the point....  Remember in the original questions, there were four possible answers.

 The first was Linear for linear equations.  This would be an equations similar to f(x) = 3x + b where b is some constant value.  Is there an equation like this that would work for the rabbits?   It's obvious (I hope) that there's a 3 involved somewhere.  All of the rabbits totals are divisible by 3 (except the first one).  What would the linear equation be if there were 0 rabbits?  f(0) = 3(0) + b, so b would have to equal 1.  Does the equation work for f(1)?  f(1) = 3(1) + 1 = 4.

The quadratic form of the equation would be f(x) = ax2 + bx + c. Wow, we could spend some time here. But this isn't the right answer either. There are no values of a, b, and c that would generate the number of rabbits for that number of months. The third answer is of the form f(x) = ax + b.

Whew - that looks complex.  But we're really approaching this backwards.  Look at the data.  At 0 months how many rabbits do we have?  Remember that any number to the power of 0 equals 1.  That's encouraging.  How many rabbits in the first month?  3 - so if a = 3 then 3 to the first power is 3 - that works.  Our equation becomes a fairly simple ...

f(x) = 3x

So, we can see that the number of rabbits increases exponentially!

Still confused?  It would have only taken a few seconds to graph this data.

I think it shows that this graph did not take me much time to create.  Yet it demonstrates and extreme increase in number of rabbits each month.  You might say that the growth rate is exponential!

Answer D?  Absolute Value -  I'm hoping no one used that answer  if it were an absolute value equation, then we'd also produce three rabbits at negative one month.  

So, the correct answer was C.  And yes, I have noticed that a disproportionate number of answers seem to be C.  Don't count on that to help you with a real test.  There are several good apps that would have given you really pretty graphs if you would like to download them to your phone.

The new question!!!!!!

Oh, the excitement is palpable isn't it?  (I just wanted to use the word palpable).  Let's do it this way.  At the end of the first month, we have produced 3 figurines of Bob.  At the end of the second month there are 6 figurines.  At the end of the third month there are 9,  Finally, at the end of the fourth month we have 12 figurines.

What type of function would represent this data?

a)  Linear

b)  Quadratic

c)  Exponential

d) Absolute Value

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