Our first objective is to try to find a pattern linking:
1. the inputs
2. the outputs
We know from the question that there are 2 inputs that make up each function and that it outputs 1 value. Hence, our first step should be trying out different inputs.
For simplicity, we will keep 1 value constant whilst changing the other value. We will start by keeping the number '1' constant. This is because '1' is an element of almost all sets and hence if the inputs are linked by any particular pattern, inputting the value '1' will not affect the result. Then, we can change the other value by adding '1' to it each time.
[Test 1: Input = 1, 1 Output = 0
Test 2: Input = 1, 2 Output = 1
Test 3: Input = 1, 3 Output = 1.58...
Test 4: Input = 1, 4 Output = 2]
Now, repeat this with a different constant.
[Test 5: Input = 2, 2 Output = 2
Test 6: Input = 2, 3 Output = 2.58...
Test 7: Input = 2, 4 Output = 3
Test 8: Input = 2, 5 Output = 3.32...]
From these few tests, we can see that the outputs are either whole numbers or decimals. Because it is hard to identify patterns from decimal outputs, we can ignore those tests and only focus on the tests that outputs whole numbers.
To achieve our first objective, we will need to compare the inputs that produces whole numbers and list out any possible relationships.
List of inputs that outputs whole numbers:
1 & 1 1 & 2 1 & 4 2 & 2 2 & 4
Possible relationship(s):
- Powers of 2 (1 = 2^0, 2 = 2^1...)
Try inputting the powers of 2 and see if you can find a pattern.
[Test 9: Input = 2, 16 Output = 5
Test 10: Input = 4, 64 Output = 8
Test 11: Input = 8, 8 Output = 6
Test 12: Input = 1, 32 Output = 5]
Now we can confirm that the inputs that produce integer outputs are in the set {powers of 2}.
Because we know that whole number inputs can produce either integer or decimal outputs, we can determine the fact that the function is an inverse function (i.e. not multiplication, addition, squaring...).
E.g.
If x is a whole number,
f(x) = 2x,
then f(x) must equal to a whole number
Else if x is a whole number,
f(x) = x^7
then f(x) must equal to a whole number
And now we can find the function based on the fact that all the inputs that produce integer outputs are powers of 2; we can infer that the function is log base 2 of x as this function will only produce integer outputs if the input is a power of 2 - which fits our current situation.
Now, assume the function is log base 2 of x, write out both the inputs and outputs.
[Test 9.1: Input = 2, 16 Output = 1, 4
Test 10.1: Input = 4, 64 Output = 2, 6
Test 11.1: Input = 8, 8 Output = 3, 3
Test 12.1: Input = 1, 32 Output = 0, 5]
Now, use the outputs for f(x) = log base 2 of x as the inputs and the original output as the output.
[Test 9.2: Input = 1, 4 Output = 5
Test 10.2: Input = 2, 6 Output = 8
Test 11.2: Input = 3, 3 Output = 6
Test 12.2: Input = 0, 5 Output = 5]
From here, we can see that the function is X1 + X2.
So if we put together both functions, we get:
f(x) = log base 2 of a + log base 2 of b
**a = first input, b = second input