We want to find a rule for a nonlinear function such that y is negative for x=1, positive for x=2, negative again for x=3, positive at x=4, and so on. Note that we can obtain this behavior when multiplying a series of negative numbers.
- 2 (-2) &= 4
-2 (-2) (-2) &= - 8
-2 (-2) (-2) (-2) &= 16
This is because multiplying two negative numbers gives us a positive result, and the product of a positive and a negative number gives us a negative number once again. We can use this for our function. In the example shown below, we can see that we will obtain this by using a power where the base is a negative number.
rcccc
(-2)^2 &=& 4
(-2)^3 &=& - 8
(-2)^4 &=& 16
For simplicity, we can use - 1 as the base. If we want the exponent to be changing, we can set it as the independent variable. Let's try with the function y = (- 1)^x.
x
(- 1)^x
y
1
(- 1)^1
- 1
2
(- 1)^2
1
3
(- 1)^3
- 1
4
(- 1)^4
1
Looking at the last column of the table, we can see that we found the behavior we wanted. The function y = (- 1)^x is negative for x=1, positive for x=2, negative for x=3, and positive at x=4, as required.
Note that there are infinitely many solutions that satisfy the condition. This is just one of them.
