How to obtain and graph a function that first grows exponentially and then decays exponentially?

Question

I would like to know what the equation of a curve is if it grows exponentially (let's say it doubles each time). This would be:

$f(x)=2^x$

But then I would like the line to exponentially decay (let it half each time once it reaches let's say 7) with an asymptote of 10.

How could I write this equation? I am only a sophomore in high school right now taking Pre-IB Algebra II by the way.

Answer 1

Since your curve does two very different things on the two intervals, you will need to define a piecewise function.

When $x \leq 7$, your function would be $f(x)$ = $2^x$

When $x > 7$, your function would be $f(x)$ = ${0.5}^x + 10$.