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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