Question

Describe the discontinuties...
See below...

Answer 1

@Hero

Answer 2

\[f(x)=(x^2+x-6/)(x^2-5x+6)\]

Answer 3

find an equivalent form for f that shows how the graph of f is related to the graph y=1/x.

Answer 4

First factor the top and bottom

Answer 5

okay. give me a second

Answer 6

(x+3)(x-2)/(x-2)(x-3)

Answer 7

@Hero okay i got it.

Answer 8

how do you find the other point of discontinuity equation?

Answer 9

Just so you know:

It is correct to express the factored form as

f(x) = ((x+3)(x-2))/((x-2)(x-3))

Answer 10

* i meant would you graph the equation

Answer 11

The discontinuity will depend on the denominator

Answer 12

now how do i find an equivalent form for f that shows how the graph of f is related to the graph y=1/x.

Answer 13

There will be no discontinuity at 2?

Answer 14

Observe:
https://www.desmos.com/calculator/a0cxbwsuep

Answer 15

okay, what about the next step?

Answer 16

Hang on

Answer 17

Okay, I have it

Answer 18

Actually it is pretty interesting that they would give you this question.

Answer 19

Remember, we factored

\(f(x) = \dfrac{(x - 2)(x + 3)}{(x - 2)(x - 3)}\) to get \(f(x) = \dfrac{x + 3}{x - 3}\)

Answer 20

From here, we can do a little trick to put \(\dfrac{x + 3}{x - 3}\) in mixed form

Answer 21

Here's how to do that:


\(\dfrac{x + 3}{x - 3} = \dfrac{x - 3 + 6}{x - 3} = \dfrac{x - 3}{x - 3} + \dfrac{6}{x - 3}\)

Answer 22

Now notice that \(\dfrac{x - 3}{x - 3} = 1\) so

\(f(x) = 1 + \dfrac{6}{x - 3}\)

Answer 23

We can rewrite it as


\(f(x) = \dfrac{6}{x - 3} + 1\)

Or even as

\(f(x) = 6\left(\dfrac{1}{x - 3}\right) + 1\)

Answer 24

So in essence, f(x) is the graph of y shifted to the right by 3, vertically stretched by 6, and vertically shifted up by 1 unit.

Answer 25

Here are both graphs plotted simultaneously which demonstrates their relationship:
https://www.desmos.com/calculator/rfgl4ey9wt

Answer 26

This is with f(x) expressed in the alternative form:
https://www.desmos.com/calculator/cvnrvmaame

Answer 27

y = (x+ 3)(x - 2)/(x - 3)(x - 2)
There is discontinuity at x = 2. y can be simplified if x is not 2.
y = (x + 3)/(x - 3)
Left side: The graph is a parabola that discontinues at x = 2
y = 0 -> x = -3
x - 0 -> y = -1
x -> +infinity -> y -> +infinity
Right side
x = 3, there is an asymptote at x = 3. y -> +infinity
x = 5 -> y = 4
x = 7 -> y = 10/4
x -> +infinity, y -> 1, since lim. y --> 1.

Answer 28

The graph of y would likely be the one in the draw.

Answer 29

@thu1935, I agree with what you said. I got discontinuity and asymptote mixed up, but your graph is wrong. Take a look at this:
https://www.desmos.com/calculator/weec7po39v

Answer 30

Yes, your graph is correct.
I made a mistake. When x -> -infinity, y goes to lim. y = 1.