Function that satisfies the given (x,y) values I am trying to come up with a function that (approximately) satisfies these $(x,y)$ values.
$(2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 6), (9, 5), (10, 4), (11, 3), (12, 2), (13, 1), (14, 2), (15, 3), (16, 4), (17, 5), (18, 6), (19, 7), (20, 8), (21, 7), (22, 6), (23, 5), (24, 4), (25, 3), (26, 2), (27, 3), (28, 4), (29, 5), (30, 6), (31, 7), (32, 8), \text {(other, anything)}$
I want to create a continuous function $f(x)$ which takes $x$ as input and outputs the corresponding $y$ value as given above. If possible, I don't want a piecewise function.
Is there a technique to do this?
 A: This function isn't exactly pretty, and it's not in the form $y=f(x)$ (I'm not even sure it's possible to put it in that form), but at the very least, it's continuous and passes through all the points you specified.
$$\frac{y-\frac{x}{13}}{\left(\sqrt{y-\frac{x}{13}}\right)^2}(7-|x-7|-y)(8-|x-20|-y)(9-|x-33|-y)=0$$
I made it by first finding the equation of the first peak, which is $y=7-|x-7|$, and then superimposing it with the graphs of the other two peaks; that's what the three sets of parentheses are for. The fraction in front limits the range of the graph to all points where $y\ge x/13$, since $y=x/13$ is the line connecting the two lowest points you provided. Here's what it looks like (with the function in blue and your points in green):

Interesting sidenote/tangent: if you wanted to generalize that function to continue the pattern that your points make, this should do the trick, where $i$ and $j$ are the $y$-coordinates of the first and last peaks, respectively:
$$\frac{y-\frac{x}{13}}{\left(\sqrt{y-\frac{x}{13}}\right)^2}\prod_{n=i}^j(n-|x-13n+84|-y)=0$$
It's equivalent to the above function and graph when $i=7$ and $j=9$. For example, here's what it looks like when $i=4$ and $j=12$:

Here's a link to the Desmos page I used to make those functions: https://www.desmos.com/calculator/8vcxwsaebq
