# How to come up with formula for this number growth

I have difficulties to come up with a formula to achieve this in my programming challenge :

• The input number ranges between 100 to 10.000 (all integers), then the output would be 0.1 - 10, so the formula is (input / 1000)
• But as the number increases, the formula also changes. From the range of 10.000 - 100k, the output would be 10 - 20. The input of 20.000 will result in 11, 30.000 will result in 12, 25.000 will result in 11.5. This is where i get lost, as i dont know how to come up with the formula to get the results based on the input in this range.

My question would be :

• As somebody that is getting interested in learning math again, what 'type' of math is it for this kind of problem ? Algebra ?
• How do i come up with the formula ? I've tried dividing the inputs with the output i expect, but the growth of the number isnt 'predictable', like a list of growing numbers with a formula. Here's what i've come up with with the format of : 10k(1k), 20k(1.818k), 30k(2.5lk), 40k(3.076k).

And i apologize for the 'approximate' tags i can come up with at this level.

It looks like $f(x)=\begin {cases} x/1,000 & x \le 10,000 \\ x/10,000+9 & x \gt 10,000 \end {cases}$
but that results in $f(100,000)=19$, not $20$ There is nothing wrong with a piecewise definition. To be a function, it needs to return a single value for any input in the domain. This one is continuous, but that is not a requirement for being a function.