Continuity (piecewise-defined function)

[Becky4paws]

I need some additional references on the example problem below. I'm confused about these types of problems and would like to know if you have a 'study reference' guide to help me understand them better before I try to actually solve it.

f(x) = Ax - 3 if x <2
= 3 - x +2x^2 if x > or = 2

I understand that Ax - 3 approaches x from the left and 3-x+2x^2 approaches x from the right but I don't understand much more about these and would like extra study material.

 

[skeeter]

a function is continuous at x = c if

1. f(x) is defined at x = c

2. lim{x->c} exists, that is lim{x->c^-} f(x) = lim{x->c⁺} f(x)

3. lim{x->c} = f(c)


for your problem, c = 2

1. f(2) is defined

2. A(2) - 3 = 3 - (2) + 2(2)²

what value of A makes this equation work?

3. now, does lim{x->2} f(x) = f(2) ?

 

[Becky4paws]

Still Missing Something ...

I guess I'm having a problem with the definitions I'm given. Maybe you can clear it up a bit.

A function f(x) is said to be continuous on an open interval (I understand open interval) a<x<b if it is continuous at each x in the interval. (Don't understand where the a and b come from). It is said to be continuous on a closed interval a<=x<=b if f(x) is continuous on the open interval a<x<b and (here's where I just get lost) f(x) approaches f(a) as x approaches a from the right (for a<x) and f(x) approaches f(b) as x approaches b from the left (for x<b).

Maybe if it could be explained in simplier terms...?

 

[stapel]

What are the instructions for this piecewise function? What are you supposed to be doing with it?

Thank you.

Eliz.

 

[Becky4paws]

Reply

The instructions read: find the values of the constant A such that the function f(x) will be continuous for all x (on the previous problem).

Would you rather me to give you the example problem from the book and maybe you can explain why the steps were done? I truly don't understand this enough to try and work the problems.

 

[stapel]

If you just need to find continuity (connectedness), then evaluate the pieces at the endpoint value, set the resulting expressions equal, and solve.

. . . . .A(2) - 3 = 2A - 3
. . . . .3 - (2) + 2(2)² = 3 - 2 + 8 = 9

What does A need to be so 2A - 3 = 9?

Eliz.