Unformatted text preview: axis. 362 Exponential and Logarithmic Functions (+) 0 (−) 0 (+)
−1 4
y = r(x) = 2x 2 −3x − 16 x e
2. The ﬁrst step we need to take to solve ex −4 ≤ 3 is to get 0 on one side of the inequality. To
that end, we subtract 3 from both sides and get a common denominator ex
ex − 4 ≤3 ex
−3 ≤ 0
−4
3 (ex − 4)
ex
−
≤ 0 Common denomintors.
ex − 4
ex − 4
12 − 2ex
≤0
ex − 4
ex x −2
We set r(x) = 12x −e and we note that r is undeﬁned when its denominator ex − 4 = 0, or
e4
when ex = 4. Solving this gives x = ln(4), so the domain of r is (−∞, ln(4)) ∪ (ln(4), ∞). To
ﬁnd the zeros of r, we solve r(x) = 0 and obtain 12 − 2ex = 0. Solving for ex , we ﬁnd ex = 6,
or x = ln(6). When we build our sign diagram, ﬁnding test values may be a little tricky since
we need to check values around ln(4) and ln(6). Recall that the function ln(x) is increasing4
which means ln(3) < ln(4) < ln(5) < ln(6) < ln(7). While the prospect of determining the
sign of r (ln(3)) may be very unsettling, remember that eln(3) = 3, so r (ln(3)) = 12 − 2eln(3)
12 − 2(3)
= −6
=
ln(3...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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