Functions and Combinations of Functions
Many functions are continuous at every real number, x. These functions include (but are not limited to):
- all polynomials (including lines)
- e^{x }
- sin(x) and cos(x)
It's helpful to see the continuity by graphing the functions. If we graph any of the above functions, we see a nice smooth graph that continues across the whole x-axis, with no jumps or holes. Try it; it will bring you and your TI-83 closer together.
Many other functions are continuous everywhere that they're defined, including
- ln(x): continuous for all x > 0
- tan(x): continuous in between multiples of and
- all rational functions that don't have common roots in the numerator and denominator: these will have vertical asymptotes at the roots of the denominator, and be continuous in between those asymptotes.
Once we know a couple of functions that are continuous at a point c, we can build other functions that are continuous at c by combining the functions we already have. To do this, we use some properties of limits.
If f and g are continuous at c, then
1. We can add or subtract:
(f + g) and (f – g) are continuous at c.
2. We can multiply:
(fg) is continuous at c.
3. We can divide functions:
is continuous at c as long as g(c) ≠ 0.
4. We can compose:
The composition (f ο g) is continuous at c.
All that's required here is that we have two functions continuous at c. It doesn't matter which is f and which is g. By switching f and g in our minds, we also find that (g – f) is continuous at c, g ο f is continuous at c, etc.
Sample Problem
Let f(x) = x + 1 and g(x) = e^{x}. These functions are both continuous at every real number x. The following functions are also continuous at every real number x:
1. We can add or subtract:
(f + g)(x) = x + 1 + e^{x} (which is the same as (g + f)(x))
(f – g)(x) = (x + 1) – e^{x}
(g – f)(x) = e^{x }– (x + 1) = e^{x }– x – 1
2. We can multiply:
(fg)(x) = (x + 1)(e^{x}) = xe^{x} + e^{x} (which is the same as (gf)(x))
3. We can divide:
(This is continuous at every real number since e^{x} is never 0.)
4. We can compose:
(f ο g)(x) = (e^{x}) + 1
(g ο f)(x) = e^{x + 1}
Also, the function is continuous at every real number except x = -1.