study the continuity of a function

[Dontknow462]

sqrt(2/3-x)^2 if x>2/3
(2-3x)/(x-1) if x equal or < 2/3

1)study the continuity of function
2) study first derivative
3) determine equation of vertical and not vertical Asymptote if they exist


pls explain me step by step, this is really important to me or I'm going to fail.

 

[tkhunny]

Does it help to observe that [math]\left(\sqrt{2/3 - x}\right)^{2} = |2/3 - x|[/math]?
Can we narrow things down even more by observing x > 2/3?

 

[topsquark]

sqrt(2/3-x)^2 if x>2/3

Is this a typo? It doesn't make a lot of sense to me. The combination of the square root and squared operations seem off.

-Dan

 

[pka]

Is this a typo? It doesn't make a lot of sense to me. The combination of the square root and squared operations seem off.

Don't think so. See reply #2 by Tkh.
\(\displaystyle f(x)=\begin{cases}\left|\frac{2}{3}-x\right| &: x>\frac{2}{3} \\ \frac{2-3x}{x+1} &: x\le \frac{2}{3}\end{cases}\)

 

[topsquark]

I did. I just think it looks like it might be a typo.

-Dan

 

[pka]

I did. I just think it looks like it might be a typo.

What is the typo?

 

[topsquark]

I couldn't tell you. But I've never seen a problem given (aside from introductory exponent manipulation) such that we are using [math]\left ( \sqrt{x} \right ) ^2[/math] at the outset. I'm not saying it's wrong, I'm just a bit surprised that someone would write a function like that.

-Dan

 

[Steven G]

I couldn't tell you. But I've never seen a problem given (aside from introductory exponent manipulation) such that we are using [math]\left ( \sqrt{x} \right ) ^2[/math] at the outset. I'm not saying it's wrong, I'm just a bit surprised that someone would write a function like that.

-Dan

It's common in the math field that [math]\left ( \sqrt{x} \right ) ^2 = |x|[/math]. Physicists!

 

[topsquark]

I know that [math]\left ( \sqrt{x} \right )^2 = |x|[/math]. Let's try an analogy. Would you give an 11th grade Math student a problem that has the problem statement: Solve for x: x + 2x = 3? It's a valid problem but looks a little funny at that level.

-Dan

 

[Steven G]

I know that [math]\left ( \sqrt{x} \right )^2 = |x|[/math]. Let's try an analogy. Would you give an 11th grade Math student a problem that has the problem statement: Solve for x: x + 2x = 3? It's a valid problem but looks a little funny at that level.

-Dan

I am sure that you know that [math]\left ( \sqrt{x} \right )^2 = |x|[/math] but seemed to be surprised to see it. Hence the comment --Physicist.

 

[topsquark]

Naw. I know it, I just rarely use it!

Note, though, I'm not the only one who doesn't use it. See here.

-Dan