What is the function rule for input x 1,2,3,4,5,6 and output 0,1,3,6,10,15?
Thanks
To find the function rule for the given input-output pairs, we need to examine the pattern in the outputs based on the corresponding inputs.
Looking at the outputs, we can observe that each output is obtained by adding the input value to the sum of all previous input values. Let's break this down:
For the first output, 0, there are no previous input values to consider, so it remains 0.
For the second output, 1, it is calculated by adding the second input (2) to the sum of the previous input (1). Therefore, 1 = 0 + 2.
For the third output, 3, it is calculated by adding the third input (3) to the sum of the previous inputs (1 + 2 = 3).
For the fourth output, 6, it is calculated by adding the fourth input (4) to the sum of the previous inputs (1 + 2 + 3 = 6).
For the fifth output, 10, it is calculated by adding the fifth input (5) to the sum of the previous inputs (1 + 2 + 3 + 4 = 10).
Finally, for the last output, 15, it is calculated by adding the sixth input (6) to the sum of the previous inputs (1 + 2 + 3 + 4 + 5 = 15).
Based on this pattern, we can conclude that the function rule for this input-output table is a quadratic relationship, where the output (y) is given by the sum of all previous inputs, up to the current input (x). Thus, the function rule can be expressed as:
y = 1 + 2 + 3 + ... + (x-1) + x
Alternatively, we can simplify this rule:
y = x(x+1)/2
Therefore, the function rule for the given input-output pairs is y = x(x+1)/2.