Equation for x, y plotting of a "dog leg"?

Question

I looking for something that be close to that basically y = "something(x)" I understand I can offset for the first part that is straight. I was looking for something like a dog leg that drops off quickly and approaches the line slowly?

It could be simple or even close.

My crude drawing below ;)

Answer 1

Let y = 1/2 if x < 1 and y = 2^-x, if x ≥ 1.

The graph has a "dog leg" and approaches the x-axis as x gets very large.