Question

I have a question regarding Limits & Continuity of functions. The problem I'm stuck on is...

"For what values of the constant "c" is the function "f" continuous on (-∞ ,∞ )?
f(x) = [cx^2 + 2x, if x<2] and [x^3 - cx, if x ≥ 2].

Any and all help is greatly appreciated!

Answer 1

Both parts of f(x) are polynomials, so you only need to make sure they have the same value for f(2). Plug in 2 for x and set the two parts equal to solve for c

4c + 4 = 8 - 2c

Answer 2

The first function isn't continuous at x=2 though, correct? The second one is continuous from the right?
Also, so we solve for "c" by setting the polynomials equal to each other and plugging in the 'conditional' values, and that is the answer to the problem? The "values of the function where "f" is continuous from (-∞ ,∞ )? It's that simple? :)

Answer 3

Sorry if my responses are slow... dorm wifi is crapping out here :p

Answer 4

sorry I'm also having trouble on my end

Answer 5

What I meant was that the functions are both continuous at x = 2 when they aren't a part of a piecewise function, so there are no other discontinuities to consider except the one at x = 2.

Just plug in 2 for x to solve for c. If you need to prove continuity, you'd take the limits on each side and they'd have to equal f(2)