Explore BrainMass

# Over what intervals are the following functions continuous?

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Please see attached file for full problem description.

Over what intervals are the following functions continuous? Justify your answer using the definition of continuity.

a.
b.

Let f and g be twice differentiable functions such that for all x in the domain of f. If and What (if anything) can you say about f, g, and h at x = 2?

https://brainmass.com/math/graphs-and-functions/over-intervals-following-functions-continuous-148496

## SOLUTION This solution is FREE courtesy of BrainMass!

a), h(x) is continuous for all real x values based on the fact that superposition of elementary functions is continuous function everywhere where the components exist and are continuous functions (we must consider domains for all the components). x^2 exists for all real x, sin(x) exists for all real x, and these elementary functions are continuous. Thus sin(x^2) is continuous function for all real x.

Similar situation for k(x). Square root of (x^2 + 7) exists for all x because (x^2 + 7) > 0. Division does not exists when denominator is equal to zero. This corresponds to x = +/- 3 ( for these x values x^2 +7 = 16 and denominator is zero).
Thus k(x) is continuous everywhere except x = +/- 3.

h'(x) = f ' (g(x)) * g ' (x) . h ' (2) = 4, so h ' (2) > 0. Because f ' (x) <= 0 we must have g ' (2) < 0. Thus, the result is the following.
g(x) is decreasing function at x = 2, f(x) is decreasing function at x = 2, but h(x) is increasing function at x = 2.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!