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August 15, 2013 Math 2254 sec 010
Section 6.1: Inverse Functions
Definition: A function f is called one-to-one if
x1 6= x2 implies f (x1) 6= f (x2).
A one-to-one function can’t take on the same value twice!
e.g. f (x) = x2 is NOT one-to-one because f (1) = f (−1) even though1 6= −1.
e.g. f (x) = x3 IS one-to-one because the only way x31 = x3
2 is ifx1 = x2.
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Horizontal Line Test
A function f is one-to-one if and only if no horizontal line intersects thegraph y = f (x) more than one.
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Graphically: One to one functions pass the HorizontalLine Test
Figure : A function that is NOT 1:1 (left) and one that is 1:1 (right).
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Inverse Functions
Definition: Let f be a one-to-one function with domain A and range B.Then f has an inverse function f−1(x) defined by
f−1(y) = x if and only if f (x) = y .
Remarks: (1) The domain of f−1 is B (the range of f ).(2) The range of f−1 is A (the domain of f ).(3) The notation f−1 does not mean ”reciprocal”.(4) The following relationships hold:
f(
f−1(x))= x for each x in A, and
f−1 (f (x)) = x for each x in B.
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ExampleFind the inverse function f−1 and verify the properties in (4), for
f (x) =√
x − 2, where A = {x |x ≥ 2}, B = {y |y ≥ 0}.
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Figure : The graph of f−1 is the reflection of the graph of f in the line y = x .
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Continuity and DifferentiabilityTheorem: If f is one-to-one and continuous on an interval, then itsinverse f−1 is continuous.
Inverses of continuous functions are continuous.
Next: Let f be 1:1 with inverse f−1 and suppose
f (b) = a i.e. f−1(a) = b.
Theorem: If f is differentiable at b, and f ′(b) 6= 0, then f−1 isdifferentiable at a and (
f−1)′
(a) =1
f ′(b).
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ExampleConsider
f (x) =√
x − 2, with f−1(x) = x2 + 2, x ≥ 0.
Evaluate the two derivatives
f ′(6), and(
f−1)′
(2).
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Plausibility ArgumentSuppose f ′(x) 6= 0, and consider the fact that f−1(f (x)) = x . Useimplicit differentiation to conclude that
ddx
f−1(f (x)) =1
f ′(x).
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Example
f (x) = x3+3 sin x+2 cos x . Find(
f−1)′
(2).
(It’s not necessarily obvious, but f is one-to-one.)
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Section 6.2*: The Natural Logarithm
We know that ∫xn dx =
xn+1
n + 1+ C
provided n 6= −1. But we have yet to know how to understand∫1x
dx .
We begin with a definite integral.
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Definition of the Natural Logarithm
Let x > 0. The natural logarithm of x is denoted and define by
ln x =
∫ x
1
1t
dt .
By the Fundamental Theorem of Calculus, we immediately get
ddx
ln x =1x.
Also, ln(1) =∫ 1
1dtt = 0.
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A Geometric Interpretation of the Nature Logarithm
Figure : ln(1) = 0, ln(x) is positive for x > 1 and ln(x) is negative for0 < x < 1.
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Properties of The Natural LogLet x and y be positive real numbers and r be a rational number.
(1) ln(xy) = ln x + ln y , (2) ln(
xy
)= ln x − ln y , and
(3) ln(x r ) = r ln x .
Example: Expand the expression ln(x3 + 2x)4 cos(2x)√
x sin x
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Properties of The Natural Log
limx→0+
ln x = −∞, and limx→∞
ln x =∞.
Figure : Plot of the natural log function.
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Definition: The number ee is the number such that ln e = 1
Figure : The number e ≈ 2.71828183
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Differentiation & Integration Rules
(1)ddx
ln u =1u
dudx
i.e.ddx
ln f (x) =f ′(x)f (x)
(2)ddx
ln |x | = 1x, and
(3)∫
dxx
= ln |x |+ C
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ExamplesEvaluate the derivative.
(a)ddx
ln(x7+2x3+2)
(b)ddx
ln(sec x)
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Evaluate
(a)∫
xx2 + 5
dx
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