The function *f*(*x*) = 4*x* + 1 is a line, therefore it's continuous everywhere. In particular, it's continuous at *x* = 5 with *f*(5) = 21. How must the *x*-values be restricted if we want to have 20.5< *f*(*x*) < 21.5?

Answer

Since we're looking at continuity near x = 5, 5 is our value of *c* and *f*(*c*) = *f*(5) = 21. Graph the function and set the window so that

0 ≤ *x* ≤ 100 ≤ *y* ≤ 42.

We do this so that (5,21) is in the center of the window. Now start narrowing the values of *x*, keeping 5 in the center, until the function goes out the sides of the graph instead of the top and bottom. Here's one possible progression, where we first bring *x* within 1 step of 5, then within 0.5 of 5, then within 0.25 of 5, and finally within 0.125 of 5:

If we restrict *x* so that 4.875 ≤ *x* ≤ 5.125, we find what we want: a picture where (5,21) is in the middle, and all the values of *f*(*x*) are between 20.5 and 21.5.