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In most other cases, though, rational√
exponents are preferred.
5
As mentioned in Section 2.2, f (x) = x2 = |x| so that absolute value is also considered an algebraic functions.
4 314 Further Topics in Functions √ f (x)
3x 3 2 − x
3x = 0
x=0
x=0
x=0 =
=
or
or
or
or 0
0
√
3 2−x=0
√
3
3
2 − x = 03
2−x=0
x=2 The zeros 0 and 2 divide the real number line into three test intervals. The sign diagram
and accompanying graph are below. Note that the intervals on which f is (+) correspond
to where the graph of f is above the x-axis, and where the graph of f is below the x-axis
we have that f is (−). The calculator suggests something mysterious happens near x = 2.
Zooming in shows the graph becomes nearly vertical there. You’ll have to wait until Calculus
to fully understand this phenomenon. (−) 0 (+) 0 (−)
0 2
y = f (x) y = f (x) near x = 2. √
2. In g (x) = 2 − 4 x + 3, we have two radicals both of which are even indexed. To satisfy
√
√
√
4
x + 3, we require x + 3 ≥ 0 or x ≥ −3. To satisfy 2 − 4 x + 3, we n...

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