Given variables and a set of results where the results, if graphed, all fall on a continuous line that has one and only one y for every x--can anyone help me come up with the formula that would use these inputs (.4 and .999 through -.999)and produce these results? Note the formula would only approach 1 and would not work if the second variable equal to or > 1 or -1.
0.4 0.999 0.91963140577134100
0.4 0.900 0.87108475929007900
0.4 0.800 0.82732683535398800
0.4 0.700 0.77396327277905700
0.4 0.600 0.70550504633038900
0.4 0.500 0.61608999638638300
0.4 0.400 0.50000000000000000
0.4 0.300 0.35555647891378000
0.4 0.200 0.19485297304232900
0.4 0.100 0.05664077746397240
0.4 0.000 0.00000000000000000
0.4 -0.100 0.05664077746397240
0.4 -0.200 0.19485297304232900
0.4 -0.300 0.35555647891378000
0.4 -0.400 0.50000000000000000
0.4 -0.500 0.61608999638638300
0.4 -0.600 0.70550504633038900
0.4 -0.700 0.77396327277905700
0.4 -0.800 0.82732683535398800
0.4 -0.900 0.87108475929007900
0.4 -0.999 0.91963140577134100
To find a formula that would produce these results, we can start by analyzing the given data.
Looking at the x-values, we can see that the variable remains constant at 0.4 for all the results. This suggests that the x-value does not directly affect the y-value.
Next, let's examine the y-values. We notice that they all fall within a certain range and seem to vary smoothly. It appears that the y-values are approaching 1 as the absolute value of the second variable increases (but does not equal or exceed 1).
Based on this observation, we can conclude that the formula should involve a function that approaches 1 as the second variable (let's call it "z") gets closer to 1 or -1.
One common mathematical function that exhibits this behavior is the hyperbolic tangent function (tanh). The tanh function approaches 1 as the input approaches infinity and approaches -1 as the input approaches negative infinity.
Given that the second variable ranges from 0.999 to -0.999, we can use the tanh function by multiplying the second variable by a scaling factor to get the desired range.
Therefore, a possible formula for generating these results could be:
y = tanh(k * z)
where y is the output, z is the second variable, and k is a scaling factor. Adjusting the value of k would allow you to fine-tune the rate at which the y values approach 1.
Note: It's important to keep in mind that this is just one possible formula that could generate the given results. There may be other functions or variations that could also produce similar results.