I am looking for examples of monotonically increasing functions f(x) which can satisfy the following constraints:
- f(x=0) should be 0 and f(x>=25) should be 1.
- I want one or more parameters in the function to control the slope of the function.
The range of x is from 0 to 50.
I was thinking of tanh functions of the form - tanh(alpha*x -beta) for monotonically increasing functions, where alpha and beta are the parameters of the function. But I don't know how to incorporate my constraint f(x=0) should be 0 and f(x>=25) should be 100.
Can you please help me with some examples of monotonically increasing functions which can satisfy my requirements?
Many thanks for your time and help!
You have several options. You can use a function like $\min$. So let's start as you did, with $f(x)=\tanh(x/c)$, where $c$ is a parameter to control the slope. We want to change this, so that $f(25)=1$. You need to multiply by a factor, so it will look like $$f(x)=\frac{\tanh(x/c)}{\tanh(25/c)}$$ This $f(x)$ is greater than $1$. So you need to change it to $1$ for $x\ge 25$. You can do this with $\min$: $$f(x)=\min\left(\frac{\tanh(x/c)}{\tanh(25/c)},1\right)$$ Alternatively, you can define the function piecewise, where you explicitly set the value to $1$ as $x\ge 25$. This one is more versatile, since you can choose any function in [0,25] that obeys your criteria. You can impose restrictions on the derivative at $0$ and $25$ for example.