Tasked with making a graph...

[Chris686]

I have to find some limits based on a graph, which is easy. I'm having issues putting the graph together though, so help would be much appreciated.



I'm thinking this isn't too difficult, but I'm not even sure where to start, and my book doesn't have any examples like this.

And just to verify I'm reading this correctly, the first portion would read, "Y is equal to 1 when x is greater than or equal to -1."

 

[mmm4444bot]

to verify I'm reading this correctly, the first portion would read, "Y is equal to 1 when x is greater than or equal to -1."

No. The symbol ≤ means less than or equal to. The symbol ≥ means greater than or equal to.

To the left of x=-1, the graph is y=1

To the right of x=1, the graph is y=x

Between -1 and 1, the graph is y=-x except at the origin

Pay close attention to ≤ versus < AND ≥ versus >, to determine which endpoints of the graph's pieces are included or not.

Check out this page, for some piecewise examples. A solid dot means the endpoint is included in the graph, and an empty dot means the endpoint is not included in the graph.

And, yes you're right; it's not difficult. Let us know, if you still need help. :cool:

 

[soroban]

Hello, Chris686!

\(\displaystyle \text{2. Given: }\:f(x) \;=\;\begin{Bmatrix}1 && x \le 1 \\ -x && -1 < x < 0 \\ -1 && x = 0 \\ -x && 0 < x < 1 \\ x && x \ge 1 \end{Bmatrix}\)
\(\displaystyle \text{Graph the function.}\)

Let's understand the pieces of this piecewise function.

If \(\displaystyle x \le 1\), the graph is the horizontal line \(\displaystyle y = 1.\)

Between \(\displaystyle \text{-}1\) and \(\displaystyle 0\), the graph is the "southeast line" \(\displaystyle (\searrow)\).

At \(\displaystyle x = 0\), the graph is the point \(\displaystyle (0,\text{-}1)\)

Between \(\displaystyle 0\) and \(\displaystyle 1\), the graph is the "southeast line".

If \(\displaystyle x \ge 1\), the graph is the "northeast line" \(\displaystyle (\nearrow)\).


The graph looks like this:

                    |
                    |           *
                    |         *
                    |       *
        * * * *    1+     *
                *   |
                  * |
      - - - - + - - o - - + - - - - - - -
             -1     | *   1
                    |   *
                    *     o
                    |

 

[Chris686]

Thanks a lot for the help. I've got it now. This is one of those cases where seeing it verbally actually helps.

 

[Chris686]

And one last thing, because I'm nervous about this test

If someone would please verify my answers. (This is in reference to the previous graph.)



a) limit is 1
b) limit is 1
c) limit is 1
d) limit is -1
e) limit is -1
f) limit is -1

part 2
(a)
The function f is continuous at -1 because of the piece 1, x =< -1

(b)
The function is not continuous at 0, because the point (0, -1) breaks the line.

 

[tkhunny]

No
Yes
No
No
No
No

You might need to review the section. "Approaching" is not "at". Very important.

 

[mmm4444bot]

part 2

(a)
The function f is continuous at -1 because of the piece 1, x =< -1

(b)
The function is not continuous at 0, because the point (0, -1) breaks the line.

Your answers for 2(a) and 2(b) are correct, but your stated reasons why are not quite adequate. Try to explain these using the definition of continuity.

For example, if we were to state why the graph is not continuous at x = 1, we could say something like the following:

The limit as x approaches 1 from the left does not equal the limit as x approaches 1 from the right.

This is a good explanation because the definition of continuity tells us that the limits from each direction need to be equal at any point of continuity.

In other words, as we move along the graph toward x=1 from the LEFT we are heading toward the y-value -1, but as we move along the graph toward x=1 from the RIGHT we are heading toward the y-value +1. Hence, the limit as x approaches from the left is -1, but the limit as x approaches from the RIGHT is +1. These limits are unequal, so the function is not continuous at x=1.