# Question 530ce

Aug 17, 2017

Well, for algebra, a point of discontinuity is a point where the function is not defined. (not entirely 100% true - you'll learn a more complete definition of discontinuity if you take calculus, which I hope you do because it opens all kinds of doors for you).

So, where might that be?

Well, what can't you divide by?

You can't divide by ZERO, right?

The divisor is x + 3.

x + 3 = 0 where x = -3.

So, label THAT point: x = -3. (The problem statement is cut off in the picture, I'm guessing it says "label the points of discontinuity..."

GOOD LUCK

Okay, I'm informed that the part of the problem that was cut off was
"Simplify the function, sketch, and ..."
...we've already labeled the point of discontinuity.

How can we SIMPLIFY the function?

It would be nice if we could factor out the (x+3) on the bottom.

What is there that would multiply by (x+3) to give the numerator?
(3x^2 + 5x - 12)

It's got to have a minus sign, x minus something.
It needs to have 3x in it. So,
(x+3) (3x - ?) = 3x^2 + 5x - 12#
hmm. $\left(3\right) \left(- 4\right) = - 12$. Does that work?

$\left(x + 3\right) \left(3 x - 4\right) = 3 {x}^{2} - 4 x + 9 x - 12 = 3 {x}^{2} + 5 x - 12$

so you can rewrite the original equation:

$\frac{3 {x}^{2} + 5 x - 12}{x + 3} = \frac{\left(x + 3\right) \left(3 x - 4\right)}{x + 3}$

and now the (x+3) terms factor out.

So then f(x) = 3x - 4

this is a line, with slope 3, and y intercept -4.

the x intercept is 4/3.

I bet you can graph this!

Note that when we simplify it, the discontinuity goes away!