Transcript 3.2 Inverse Functions and Logarithms 3.3 Derivatives of Logarithmic and Exponential functions One-to-one functions • Definition: A function f is called a one-to-one function if.

3.2 Inverse Functions and
Logarithms
3.3 Derivatives of Logarithmic
and Exponential functions
One-to-one functions
• Definition: A function f is called a one-to-one
function if it never takes on the same value
twice; that is
f(x1) ≠ f(x2) whenever x1 ≠ x2.
• Horizontal line test: A function f is one-to-one if
and only if no horizontal line intersects its graph
more than once.
• Examples: f(x) = x3 is one-to-one
but f(x) = x2 is not.
Inverse functions
• Definition: Let f be a one-to-one function with
domain A and range B. Then the inverse
function f -1 has domain B and range A and is
defined by
f 1 ( y)  x  f ( x)  y
for any y in B.
• Note: f -1(x) does not mean 1 / f(x) .
• Example: The inverse of f(x) = x3 is f -1(x)=x1/3
• Cancellation equations:
f 1 ( f ( x))  x for every x in A
f ( f 1 ( x))  x for every x in B
How to find the inverse function of a
one-to-one function f
• Step 1: Write y=f(x)
• Step 2: Solve this equation for x in terms of y (if
possible)
• Step 3: To express f -1 as a function of x, interchange
x and y. The resulting equation is y = f -1(x)
• Example: Find the inverse of f(x) = 5 - x3
y  5 x
3
x  5 y x  3 5 y
3
T hus, theinversefunctionis f ( x)  3 5  x
Another example:
1
f  x  x 1
2
5
1
y  x 1
2
4
3
2
1
Solve for x:
-5 -4 -3 -2 -1 0
-1
1
y 1  x
2
-2
-3
-4
2y  2  x
-5
1
2
3
4
Inverse functions
are reflections
about y = x.
x  2y  2
Switch x and y:
y  2x  2
5
f 1  x   2x  2
Derivative of inverse function
First consider an example:
f  x  x
df
 2x
dx
At x = 2:
2
x0
y
4
 2, 4 
Slopes are
reciprocals.
m4
2
df
 2  2  2  4
dx
df 1
1

dx
2 x
y  x2
6
f  2  22  4
f 1  x   x
8
 4, 2  m 
1
4
4
6
0
0
2
y x
8
x
At x = 4:
f 1  4  4  2
df 1
1
1
1


 4 
dx
2 4 22 4
Calculus of inverse functions
• Theorem: If f is a one-to-one continuous function
defined on an interval then its inverse function f -1 is
also continuous.
• Theorem: If f is a one-to-one differentiable function
with inverse function f -1 and f ′ (f -1 (a)) ≠ 0, then the
inverse function is differentiable and
( f )(a) 
1
1
1

f ( f (a))
• Example: Find (f -1 )′ (1) for f(x) = x3 + x + 1
Solution: By inspection f(0)=1, thus f -1(1) = 0
1
1
1
Then ( f 1 )(1) 


f ( f (1))
1
f (0)
3 0 1
2
1
Logarithmic Functions
Consider
f  x   a x where a>0 and a≠1
This is a one-to-one function, therefore it has an inverse.
The inverse is called the logarithmic function with base a.
loga x  y  a  x
y
Example:
16  24
4  log 2 16
The most commonly used bases for logs are 10: log10 x  log x
and e:
y  ln x
loge x  ln x
is called the natural logarithm function.
Properties of Logarithms
a
loga x
x
loga a  x
x
a  0 , a  1 ,
x  0
Since logs and exponentiation are inverse functions, they
“un-do” each other.
Product rule:
loga xy  loga x  loga y
Quotient rule:
x
log a  log a x  log a y
y
Power rule:
loga x  y loga x
Change of base formula:
y
ln x
log a x 
ln a
Derivatives of Logarithmic and
Exponential functions
d
1
loga x  
dx
x ln a
d x
x
a  a ln a
dx
 
Examples on the board.
d
1
ln x  
dx
x
d x
x
e e
dx
 
Logarithmic Differentiation
The calculation of derivatives of complicated functions
involving products, quotients, or powers can often be
simplified by taking logarithms.
• Step 1: Take natural logarithms of both sides of an
equation y = f (x) and use the properties of
logarithms to simplify.
• Step 2: Differentiate implicitly with respect to x
• Step 3: Solve the resulting equation for y′
Examples on the board