The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.

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
. . . . . .