Chapter Review true/false questions on page 195
#1: “If
f
is onetoone, with domain
R
, then
f
–1
(
f
(6)) = 6.”
..?..
Answer: True.
But note that
f
(
f
–1
(6)) need not be defined;
e.g., let
f
(
x
) = – exp(
x
) or exp(
x
) + 6.
#2: “If
f
is onetoone and differentiable, with domain
R
,
then (
f
–1
)
′
(6) = 1/
f
′
(6).”
..?..
Answer: False. It’s equal to 1/
f
′
(
f
–1
(6)) (assuming that the
denominator isn’t zero), and it’s easy to cook up examples
like
f
(
x
) =
x
3
where the two quantities are unequal.
#3: “The function
f
(
x
) = cos
x
, –
π
/2
≤
x
≤
π
/2, is oneto
one.”
..?..
Answer: False (note for instance that cos
π
/2 = cos –
π
/2).
But
f
(
x
) = cos
x
, 0
≤
x
≤
π
is onetoone, as is
f
(
x
) = sin
x
,
–
π
/2
≤
x
≤
π
/2.
#4: “tan
–1
(–1) = 3
π
/4”
..?..
Answer: False!
tan(3
π
/4) = –1, but since the range of tan
–1
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is (
π
/2,
π
/2) we must have tan
–1
(–1) = –
π
/4.
#5: “If 0 <
a
<
b
, then ln
a
< ln
b
.”
..?..
Answer: True, since ln
x
is an increasing function on (0,
∞
).
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 Fall '11
 Staff
 Calculus, Derivative, Inverse function, Logarithm

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