Some graphs are not defined for all values of x. By not defined, we mean there is no y-value for that x-value.

The following function is undefined at x = 3:

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_{}

When x = 3, the denominator is equal to 0, so we have:

_{}

This is undefined. Since there is a point where y is not defined, we can say that this function is discontinuous at x = 3. This means it does not have a y-value for every x-value. To plot a function like this, you need to look at a few things.

Firstly, you find where the function is undefined – in this case at x = 3. Then you work out what sort of value the function has at four points:

Approaching _{}:

_{}

The 3 and the –6 can be ignored as they are
insignificant when compared with _{}.

Approaching _{} from the negative side:

_{}

Approaching _{} from the positive side:

_{}

Approaching _{}:

_{}

Now, I can use this to plot my function near these points:

·
Towards _{}, y approaches 2. Approaching x =
3, from the negative side, y approaches a very large negative number. So when
I travel right from _{} towards x = 3, my function is
getting more negative – it is decreasing.

·
Towards _{}, y approaches 2 again. However,
approaching x = 3 from the positive side, y approaches a very large *positive* number. So if I travel left from _{} to x = 3, my
function is getting more positive – it is increasing.

With this in mind, I can draw the function around these
four points. Since I can’t draw it all the way to _{} or _{}, I just go a reasonable
distance from the y-axis.

Now it is just a simple task of joining up the lines.
Since the function is undefined at x = 3, you can draw a vertical dotted line
there – a *vertical asymptote*.

You can also draw a *horizontal* asymptote at y = 2 since y does not
actually ever get to 2.