functions breakpoints

[wizardhat112]

Task "Detect function break points , and its type, draw function graph."

I have done something but i don't know if it's right?

If its right so far then i don't understand what to do with (lim->-infinity) las part. And how to draw function.

 

[ksdhart]

Okay, so you're looking for function breakpoints, or places where the function is undefined. You've identified that the function is undefined if the denominator is zero. But are there other conditions under which some function might be undefined? Do any of those conditions apply to this specific function? And for the part where it says "..and it's type", well that's simple. What types of function break points were discussed in your book or your class? Do any of those definitions match the types of break points you found?

And next is drawing the graph of the function. You've identified that you have an asymptote when the denominator goes to zero, and looked the limit as the function approaches that x-value from either side. That's good. Now we're at the long term behaviors of the function, or examining the limits as x goes to infinity and negative infinity. Are you familiar with L'Hopital's Rule? If you are, then what do you get when you apply that to the limits? If not, let's think of another way we might examine the limits. The highest power of x in both the numerator and the denominator is just x. So, what if we divide both the numerator and denominator by x?

$\displaystyle \frac{1+x}{3-x}\cdot \frac{\frac{1}{x}}{\frac{1}{x}}=\frac{\frac{1}{x}+1}{\frac{3}{x}-1}$

Now what happens if you take the limit of this rewritten expression as x approaches infinity? As x approaches negative infinity?

 

[Ishuda]

Task "Detect function break points , and its type, draw function graph."

I have done something but i don't know if it's right?

If its right so far then i don't understand what to do with (lim->-infinity) las part. And how to draw function.

It's alright so far if I understand what you have done. So far what do you get for the break points.

For the part lim->-infinity, it would be better to take x out instead of the constants to get
$\displaystyle \underset{x\, \to\, -\infty}{lim}\, \frac{1\, +\, x}{3\, -\, x}\, =\, \underset{x\, \to\, -\infty}{lim}\, \frac{x\, (\frac{1}{x}\, +\, 1)}{x\, (\frac{3}{x}\, -\, 1)}$ = ???

 

[stapel]

Task "Detect function break points , and its type, draw function graph."

$\displaystyle y\, =\, \dfrac{1\, +\, x}{3\, -\, x}$

I have done something but i don't know if it's right?

You have found the zeroes of the numerator and the denominator, but do you know what these mean? Think back to algebra, when you were graphing rational functions. (Refresh here.) What was the point of the zeroes of the numerator? What was the (very different) point of the zeroes of the denominator?

If its right so far....

What do you mean by "the limit as x tends toward 3 + 0"? What do you mean by "the limit as x tends toward 3 - 0"? How are these different? What are you doing with the other "limits" (which don't need to be done via limits, since the function is defined at x = -1, so you can simply evaluate)? What are you trying to accomplish? Did your book (or instructor) ever do limits of this "style"?

then i don't understand what to do with (lim->-infinity) las part.

Use the limit rules, formulas, methods, and tricks they taught you. For instance, you could divide through, top and bottom, by x; then take the limit, noting that 1/x goes to zero as x gets arbitrarily large. Or you could do the long division to convert the current polynomial "fraction" to the equivalent polynomial "mixed-number" form...

. . . . .$\displaystyle \dfrac{1\, +\, x}{3\, -\, x}\, =\, -1\, +\, \dfrac{4}{3\, -\, x}$

...and then apply limits to that expression.

And how to draw function.

Do the same graphing that you learned back in algebra: Find the zeroes (that is, the x-intercepts); find the vertical asymptotes (that is, the zeroes of the denominator); find the horizontal asymptotes (using the rules you learned back then); plot some points; and then flesh in the graph.