Learning Resources
Lesson
Investigation 11
The problem in Investigation 11 is very similar to the one used in Lesson 01 of Section 01. However, in this investigation you will find two exponential functions. One will describe the total number of grains on all the squares of the chess board up to and including any given square. The other will describe the number on a given square.
The two functions are shown in two different columns of the table. One function is called f(x) and the other g(x). The number of the square on the chess board is the value of x. The table has been complete for the first five squares of the chess board.
Square # |
Number of rice grains on the square, |
Total number of rice grains on the board, |
1 |
1 |
1 |
2 |
2 |
3 |
3 |
4 |
7 |
4 |
8 |
15 |
5 |
16 |
31 |
. |
Step A of the Investigation asks you to complete the table. That is impractical for all 64 squares, but you should complete at least the next 4 or 5 rows. By then you should have seen the pattern and be able to write the function rule, f(x), asked for in Step B.
The function rule in Step C, g(x), is a little more difficult to see. There is not a common ratio between the terms, for example 
, so therefore you don't have a value for b in the basic exponential function 
.
Look closely at the values in the g(x) column. Is there some number that you can add to or subtract from each so that the resulting numbers will form a sequence that has a common difference? Use this to write the function g(x).
The following examples are similar to the work you will have to do in the Investigation. As you work through them, try to see how they are similar to the examples in the Investigation.
Example 1
For the table below, find the equation of the function p(x) that describes the numbers in the second row as a function of the numbers in the first row.
x |
1 |
2 |
3 |
4 |
5 |
.... |
y or p(x) |
1 |
3 |
9 |
27 |
81 |
... |
Solution
The desired function is exponential because the successive values for p(x) have a common ratio of 3 (e.g. 
). Thus b = 3 in the basic exponential function.
The temptation is to say that the function is p(x) = 3x , but that is not correct. To assure yourself of this evaluate p(4):
p(4) = 34 = 81
But look at the table. When x = 4 in the table, p(x) is only 27 (which is 33). In fact, all the values of p(x) are 3 to an exponent 1 less than the value of x. To see this, add another row to the table and express the values of p(x) as powers of 3.
x |
1 |
2 |
3 |
4 |
5 |
... |
y or p(x) |
1 |
3 |
9 |
27 |
81 |
|
y or p(x) |
30 |
31 |
32 |
33 |
34 |
|
Note that the exponent of the base is always 1 less than the value of x. The desired function is thus:
p(x) = 3x - 1
This is very similar to the problem posed in Step B of Investigation 11. The only difference is that there you generated the numbers from a real world application.
Example 2
For the table below, find the equation of the function h(x) that describes the numbers in the second row as a function of the numbers in the first row.
x |
1 |
2 |
3 |
4 |
5 |
.... |
y or h(x) |
1 |
7 |
25 |
79 |
241 |
... |
Solution
For this function the successive range values do not have a common ratio (for example 
). We may be tempted to think that the function h(x) is therefore not exponential in nature.
However, if you look closely at the range values of h(x) you might see that there is a pattern to them. If you add 2 to each of the range values you get the sequence 3 , 9, 27, 81, 243, .., and these are all powers of 3.
In other words, the range values of h(x) are all 2 less than the power of 3 for that value of x. This is shown by adding another row to the table and writing the values in this form.
x |
1 |
2 |
3 |
4 |
5 |
... |
y or f(x) |
1 |
7 |
25 |
79 |
241 |
... |
y or f(x) |
31 -2 |
32 -2 |
33 - 2 |
34 - 2 |
35 - 2 |
|
We can thus write the function as:
h(x) = 3x - 2
Note the difference between this and the function p(x) in the example above. For h(x) the power of 3 is found and then the number is subtracted from it. For p(x) the number was first subtracted from the domain value and then the power of 3 was found.
The method used to find h(x) should help you find the function in Step C of Investigation 11. The base and what you add or subtract may be different, but the procedure is the same.
The functions in these two examples illustrate the very important difference between functions of the following type:
f(x) = bx - k and g(x) = bx - k
Activity
- Complete Investigation 11 on page 156 in your text.
- Complete the Investigation Questions 1 - 7 on pages 156 & 157
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
Test Yourself
Find the function f(x) that describes the data in the following table and use this function to find f(10).
x |
1 |
2 |
3 |
4 |
5 |
... |
y or f(x) |
4 |
8 |
16 |
32 |
64 |
... |