Invent a number game like the one from Problem D1. Draw a network that lets you figure out the output for any number. Draw another network that lets you find someone's original number if you know his or her final result. Try out your game on someone else.
Here's a number game:
Pick a number
Double your answer
Finally, multiply by 3
Tell whether each example below is a function, and explain how you decided.
Input: a circle. Output: the ratio of the circumference to the diameter.
Input: a soccer team. Output: a member of the team.
Input: a CD. Output: a song on the CD.
Remember, a function is any relationship in which each input leads to exactly one output, and the same output may be repeated more than once for different inputs. A rule is not a function when the same input can lead to multiple outputs. Close Tip
Gabriela and Ben are trying to decide whether the rule y = x4 is a function. Represented as an algorithm, the equation y = x4 is equivalent to starting with a number, then multiplying that number by itself four times. For example:
Input: x = 2 Output: y = 2 x 2 x 2 x 2 = 16.
Who is correct, Ben or Gabriela? Is y = x4 a function? Explain how you know.
Problems H4-H7 involve iteration, a process that can be done to any function where the output is the same type as the input. When you iterate a function, you apply it again and again, each time using the previous output as the new input. Iteration is a very important technique for solving equations approximately when typical algebraic methods can't be used, and it is also used to model many real-world problems like population growth and the change of weather.
Pick a number. Add one. Add one again. Again. Again. By doing this, you're iterating the "add one" function. The "his father before him" function is frequently iterated in conversation. Close Tip
Here's an example:
Input: a real number Output: half that number
Start with an input of 20. The first output is 10. Now, use that output as the new input. The second output is 5. Use 5 as the next input, and you get an output of 2.5. And so on.
Use the function described above, starting with an input of -16. What are the first five outputs as you iterate the function? What will happen to the value if the iteration continues forever?
The first output is -8, then use -8 as the next input. Close Tip
Build a network of function machines that adds 1 to an input and then divides by 3. Now iterate the function. Try three different original inputs, and iterate the function at least 10 times for each input. What is going on?
What happens to the values "in the end," after many iterations? For what numbers does this happen? Can you explain why? Close Tip
A "fixed point" of an iteration is a value of the input that produces itself as an output. For example, if your algorithm subtracts 3 from an input, then multiplies by 2, the input value 6 has output 6, so it is a fixed point.
Find any fixed points for these algorithms. There may not be any, or there may be more than one.
Add 1, and divide by 3
Subtract 6, and multiply by 3
Square the input
Add 12, and divide by 4
Square the input, then subtract 6
If a fixed point's input is n, what would its output have to be? Close Tip
You can also iterate functions that act on geometric shapes. Here's a new function:
Start with a rightside-up triangle; that is, a triangle with one base horizontal. Find the midpoints of each side of the triangle, and connect them to each other. Repeat this process on each rightside-up triangle in the output.
Here's the start of the iteration for this function: