NOTES on undefined expressions
There are 2 main cases of undefined expressions relevant to the GED® test:
1) when you try to take the square root of a negative number

There actually are imaginary numbers that include √−1 . However, the GED® test
concerns itself only with the set of Real numbers (not imaginary ones), so any
negative number under a square root sign counts as undefined.
Consider √−25. The square root sign means you’re thinking of a number times
itself that equals –25. But there isn’t any such real number:
5 times 5 gets you positive 25, and -5 times -5 also gets you positive 25.

Note that it’s perfectly OK to have the cube root of negative numbers.
3
Consider √−64. The cube root sign means you’re thinking of a number times itself
times itself that equals –64. Well, (–4) (–4) (–4) = –64.
2) when you try to divide by zero

In the coordinate plane, a vertical line has undefined slope. Why? Because using the
slope formula would involve dividing by zero.
(x1, y1)
(x2, y2)
Consider the points (4, 3) and (4, 8)
slope =
𝑦2 −𝑦1
𝑥2 −𝑥1
=
8−3 5
=
4−4 0
The slope is undefined. (This is different from saying the slope is 0. The slope is 0 for horizontal
lines. The slope is undefined for vertical lines.)

If you see an expression with x in the denominator, find what values of x would make
the denominator 0. Those are the values of x for which the expression is undefined.
o On a graph there would be no y-value for such x-values; instead you’d often
see an asymptote, a line that the graph gets ever closer to but never reaches.
𝑥
(𝑥+3)(𝑥−8)
What would make the denominator equal to 0?
Well, if (x + 3) = 0, the whole denominator would = 0 (since 0 times anything is 0).
The same goes for (x – 8).
(x + 3) = 0 when x = – 3
(x – 8) = 0 when x = 8
When x = – 3 or when x = 8, the denominator would equal 0, so the original
expression is undefined for those values of x. The graph would get closer and closer
to those x-values but would have no y-value exactly on those x-values.
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Service LLC under license. This material is not endorsed or approved by ACE or GED Testing Service.
D. Stark 5/20/2016