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Section	3.2
                  Inverse	Functions	and	Logarithms

                               V63.0121.034, Calculus	I



                                  October	21, 2009



        Announcements
                Midterm	course	evaluations	at	the	end	of	class

        .
.
Image	credit: Roger	Smith
                                                       .   .     .   .   .   .
Outline




  Inverse	Functions



  Derivatives	of	Inverse	Functions



  Logarithmic	Functions




                                     .   .   .   .   .   .
What	is	an	inverse	function?



   Definition
   Let f be	a	function	with	domain D and	range E. The inverse of f is
   the	function f−1 defined	by:

                              f−1 (b) = a,

   where a is	chosen	so	that f(a) = b.




                                               .    .   .    .   .      .
What	is	an	inverse	function?



   Definition
   Let f be	a	function	with	domain D and	range E. The inverse of f is
   the	function f−1 defined	by:

                               f−1 (b) = a,

   where a is	chosen	so	that f(a) = b.
   So
                    f−1 (f(x)) = x,      f(f−1 (x)) = x




                                                  .       .   .   .   .   .
What	functions	are	invertible?



   In	order	for f−1 to	be	a	function, there	must	be	only	one a in D
   corresponding	to	each b in E.
       Such	a	function	is	called one-to-one
       The	graph	of	such	a	function	passes	the horizontal	line	test:
       any	horizontal	line	intersects	the	graph	in	exactly	one	point
       if	at	all.
       If f is	continuous, then f−1 is	continuous.




                                                .    .   .   .    .    .
Graphing	an	inverse	function



     The	graph	of f−1
     interchanges	the x and y               f
                                            .
     coordinate	of	every
     point	on	the	graph	of f

                                    .




                                .   .   .       .   .   .
Graphing	an	inverse	function



     The	graph	of f−1
     interchanges	the x and y                f
                                             .
     coordinate	of	every
     point	on	the	graph	of f
                                                     .−1
                                                     f
     The	result	is	that	to	get
     the	graph	of f−1 , we           .
     need	only	reflect	the
     graph	of f in	the
     diagonal	line y = x.




                                 .   .   .       .   .     .
How	to	find	the	inverse	function
 1. Write y = f(x)
 2. Solve	for x in	terms	of y
 3. To	express f−1 as	a	function	of x, interchange x and y




                                            .   .    .       .   .   .
How	to	find	the	inverse	function
 1. Write y = f(x)
 2. Solve	for x in	terms	of y
 3. To	express f−1 as	a	function	of x, interchange x and y

Example
Find	the	inverse	function	of f(x) = x3 + 1.




                                              .   .   .      .   .   .
How	to	find	the	inverse	function
 1. Write y = f(x)
 2. Solve	for x in	terms	of y
 3. To	express f−1 as	a	function	of x, interchange x and y

Example
Find	the	inverse	function	of f(x) = x3 + 1.

Answer               √
y = x3 + 1 =⇒ x =    3
                         y − 1, so
                                      √
                          f−1 (x) =   3
                                          x−1




                                                .   .   .    .   .   .
Outline




  Inverse	Functions



  Derivatives	of	Inverse	Functions



  Logarithmic	Functions




                                     .   .   .   .   .   .
derivative	of	square	root


                        √                   dy
   Recall	that	if y =       x, we	can	find      by	implicit	differentiation:
                                            dx
                              √
                        y=      x =⇒ y2 = x
                                     dy
                               =⇒ 2y    =1
                                     dx
                                     dy    1   1
                                 =⇒     =    = √
                                     dx   2y  2 x

                 d 2
   Notice 2y =      y , and y is	the	inverse	of	the	squaring	function.
                 dy




                                                      .   .    .    .    .    .
Theorem	(The	Inverse	Function	Theorem)
Let f be	differentiable	at a, and f′ (a) ̸= 0. Then f−1 is	defined	in	an
open	interval	containing b = f(a), and

                                              1
                       (f−1 )′ (b) =   ′ −1
                                       f (f   (b))




                                                     .   .   .   .   .    .
Theorem	(The	Inverse	Function	Theorem)
Let f be	differentiable	at a, and f′ (a) ̸= 0. Then f−1 is	defined	in	an
open	interval	containing b = f(a), and

                                                 1
                          (f−1 )′ (b) =   ′ −1
                                          f (f   (b))


“Proof”.
If y = f−1 (x), then
                                 f (y ) = x ,
So	by	implicit	differentiation

                      dy        dy     1         1
             f′ (y)      = 1 =⇒    = ′     = ′ −1
                      dx        dx   f (y)   f (f (x))



                                                        .   .   .   .   .   .
Outline




  Inverse	Functions



  Derivatives	of	Inverse	Functions



  Logarithmic	Functions




                                     .   .   .   .   .   .
Logarithms

  Definition
      The	base a logarithm loga x is	the	inverse	of	the	function ax

                          y = loga x ⇐⇒ x = ay

      The	natural	logarithm ln x is	the	inverse	of ex . So
      y = ln x ⇐⇒ x = ey .




                                                .    .       .   .   .   .
Logarithms

  Definition
       The	base a logarithm loga x is	the	inverse	of	the	function ax

                              y = loga x ⇐⇒ x = ay

       The	natural	logarithm ln x is	the	inverse	of ex . So
       y = ln x ⇐⇒ x = ey .

  Facts
    (i) loga (x · x′ ) = loga x + loga x′




                                                 .    .       .   .   .   .
Logarithms

  Definition
       The	base a logarithm loga x is	the	inverse	of	the	function ax

                              y = loga x ⇐⇒ x = ay

       The	natural	logarithm ln x is	the	inverse	of ex . So
       y = ln x ⇐⇒ x = ey .

  Facts
    (i) loga (x · x′ ) = loga x + loga x′
             (x)
   (ii) loga ′ = loga x − loga x′
               x



                                                 .    .       .   .   .   .
Logarithms

  Definition
        The	base a logarithm loga x is	the	inverse	of	the	function ax

                               y = loga x ⇐⇒ x = ay

        The	natural	logarithm ln x is	the	inverse	of ex . So
        y = ln x ⇐⇒ x = ey .

  Facts
     (i) loga (x · x′ ) = loga x + loga x′
              (x)
    (ii) loga ′ = loga x − loga x′
                x
   (iii) loga (xr ) = r loga x


                                                  .    .       .   .   .   .
Logarithms	convert	products	to	sums

      Suppose y = loga x and y′ = loga x′
                                  ′
      Then x = ay and x′ = ay
                   ′          ′
      So xx′ = ay ay = ay+y
      Therefore

                  loga (xx′ ) = y + y′ = loga x + loga x′




                                               .    .       .   .   .   .
Example
Write	as	a	single	logarithm: 2 ln 4 − ln 3.




                                              .   .   .   .   .   .
Example
Write	as	a	single	logarithm: 2 ln 4 − ln 3.

Solution
                                        42
    2 ln 4 − ln 3 = ln 42 − ln 3 = ln
                                        3
          ln 42
    not         !
           ln 3




                                              .   .   .   .   .   .
Example
Write	as	a	single	logarithm: 2 ln 4 − ln 3.

Solution
                                        42
    2 ln 4 − ln 3 = ln 42 − ln 3 = ln
                                        3
          ln 42
    not         !
           ln 3

Example
                                  3
Write	as	a	single	logarithm: ln     + 4 ln 2
                                  4




                                               .   .   .   .   .   .
Example
Write	as	a	single	logarithm: 2 ln 4 − ln 3.

Solution
                                        42
    2 ln 4 − ln 3 = ln 42 − ln 3 = ln
                                        3
          ln 42
    not         !
           ln 3

Example
                                  3
Write	as	a	single	logarithm: ln     + 4 ln 2
                                  4
Answer
ln 12


                                               .   .   .   .   .   .
“ .
                      . lawn”




        .




.
Image	credit: Selva
                                .   .   .   .   .   .
Graphs	of	logarithmic	functions

       y
       .
                    . = 2x
                    y


                                              y
                                              . = log2 x



       . . 0 , 1)
         (

       ..1, 0) .
       (                                           x
                                                   .




                                  .   .   .    .    .      .
Graphs	of	logarithmic	functions

       y
       .
                    . = 3x= 2x
                    y . y


                                              y
                                              . = log2 x


                                              y
                                              . = log3 x
       . . 0 , 1)
         (

       ..1, 0) .
       (                                           x
                                                   .




                                  .   .   .    .    .      .
Graphs	of	logarithmic	functions

       y
       .
             . = .10x 3x= 2x
             y y=.    y


                                              y
                                              . = log2 x


                                              y
                                              . = log3 x
       . . 0 , 1)
         (
                                              y
                                              . = log10 x
       ..1, 0) .
       (                                            x
                                                    .




                                  .   .   .     .    .      .
Graphs	of	logarithmic	functions

       y
       .
             . = .10=3xx 2x
                  y xy
             y y. = .e =


                                              y
                                              . = log2 x

                                               y
                                               . = ln x
                                              y
                                              . = log3 x
       . . 0 , 1)
         (
                                              y
                                              . = log10 x
       ..1, 0) .
       (                                            x
                                                    .




                                  .   .   .     .    .      .
Change	of	base	formula	for	exponentials

   Fact
   If a > 0 and a ̸= 1, then

                                          ln x
                               loga x =
                                          ln a




                                                 .   .   .   .   .   .
Change	of	base	formula	for	exponentials

   Fact
   If a > 0 and a ̸= 1, then

                                            ln x
                                 loga x =
                                            ln a


   Proof.
          If y = loga x, then x = ay
          So ln x = ln(ay ) = y ln a
          Therefore
                                                   ln x
                                  y = loga x =
                                                   ln a



                                                          .   .   .   .   .   .

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Lesson 15: Inverse Functions and Logarithms

  • 1. Section 3.2 Inverse Functions and Logarithms V63.0121.034, Calculus I October 21, 2009 Announcements Midterm course evaluations at the end of class . . Image credit: Roger Smith . . . . . .
  • 2. Outline Inverse Functions Derivatives of Inverse Functions Logarithmic Functions . . . . . .
  • 3. What is an inverse function? Definition Let f be a function with domain D and range E. The inverse of f is the function f−1 defined by: f−1 (b) = a, where a is chosen so that f(a) = b. . . . . . .
  • 4. What is an inverse function? Definition Let f be a function with domain D and range E. The inverse of f is the function f−1 defined by: f−1 (b) = a, where a is chosen so that f(a) = b. So f−1 (f(x)) = x, f(f−1 (x)) = x . . . . . .
  • 5. What functions are invertible? In order for f−1 to be a function, there must be only one a in D corresponding to each b in E. Such a function is called one-to-one The graph of such a function passes the horizontal line test: any horizontal line intersects the graph in exactly one point if at all. If f is continuous, then f−1 is continuous. . . . . . .
  • 6. Graphing an inverse function The graph of f−1 interchanges the x and y f . coordinate of every point on the graph of f . . . . . . .
  • 7. Graphing an inverse function The graph of f−1 interchanges the x and y f . coordinate of every point on the graph of f .−1 f The result is that to get the graph of f−1 , we . need only reflect the graph of f in the diagonal line y = x. . . . . . .
  • 8. How to find the inverse function 1. Write y = f(x) 2. Solve for x in terms of y 3. To express f−1 as a function of x, interchange x and y . . . . . .
  • 9. How to find the inverse function 1. Write y = f(x) 2. Solve for x in terms of y 3. To express f−1 as a function of x, interchange x and y Example Find the inverse function of f(x) = x3 + 1. . . . . . .
  • 10. How to find the inverse function 1. Write y = f(x) 2. Solve for x in terms of y 3. To express f−1 as a function of x, interchange x and y Example Find the inverse function of f(x) = x3 + 1. Answer √ y = x3 + 1 =⇒ x = 3 y − 1, so √ f−1 (x) = 3 x−1 . . . . . .
  • 11. Outline Inverse Functions Derivatives of Inverse Functions Logarithmic Functions . . . . . .
  • 12. derivative of square root √ dy Recall that if y = x, we can find by implicit differentiation: dx √ y= x =⇒ y2 = x dy =⇒ 2y =1 dx dy 1 1 =⇒ = = √ dx 2y 2 x d 2 Notice 2y = y , and y is the inverse of the squaring function. dy . . . . . .
  • 13. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open interval containing b = f(a), and 1 (f−1 )′ (b) = ′ −1 f (f (b)) . . . . . .
  • 14. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open interval containing b = f(a), and 1 (f−1 )′ (b) = ′ −1 f (f (b)) “Proof”. If y = f−1 (x), then f (y ) = x , So by implicit differentiation dy dy 1 1 f′ (y) = 1 =⇒ = ′ = ′ −1 dx dx f (y) f (f (x)) . . . . . .
  • 15. Outline Inverse Functions Derivatives of Inverse Functions Logarithmic Functions . . . . . .
  • 16. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . . . . . . .
  • 17. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ . . . . . .
  • 18. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ (x) (ii) loga ′ = loga x − loga x′ x . . . . . .
  • 19. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ (x) (ii) loga ′ = loga x − loga x′ x (iii) loga (xr ) = r loga x . . . . . .
  • 20. Logarithms convert products to sums Suppose y = loga x and y′ = loga x′ ′ Then x = ay and x′ = ay ′ ′ So xx′ = ay ay = ay+y Therefore loga (xx′ ) = y + y′ = loga x + loga x′ . . . . . .
  • 21. Example Write as a single logarithm: 2 ln 4 − ln 3. . . . . . .
  • 22. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 . . . . . .
  • 23. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 Example 3 Write as a single logarithm: ln + 4 ln 2 4 . . . . . .
  • 24. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 Example 3 Write as a single logarithm: ln + 4 ln 2 4 Answer ln 12 . . . . . .
  • 25. “ . . lawn” . . Image credit: Selva . . . . . .
  • 26. Graphs of logarithmic functions y . . = 2x y y . = log2 x . . 0 , 1) ( ..1, 0) . ( x . . . . . . .
  • 27. Graphs of logarithmic functions y . . = 3x= 2x y . y y . = log2 x y . = log3 x . . 0 , 1) ( ..1, 0) . ( x . . . . . . .
  • 28. Graphs of logarithmic functions y . . = .10x 3x= 2x y y=. y y . = log2 x y . = log3 x . . 0 , 1) ( y . = log10 x ..1, 0) . ( x . . . . . . .
  • 29. Graphs of logarithmic functions y . . = .10=3xx 2x y xy y y. = .e = y . = log2 x y . = ln x y . = log3 x . . 0 , 1) ( y . = log10 x ..1, 0) . ( x . . . . . . .
  • 30. Change of base formula for exponentials Fact If a > 0 and a ̸= 1, then ln x loga x = ln a . . . . . .
  • 31. Change of base formula for exponentials Fact If a > 0 and a ̸= 1, then ln x loga x = ln a Proof. If y = loga x, then x = ay So ln x = ln(ay ) = y ln a Therefore ln x y = loga x = ln a . . . . . .