Derivatives of Inverse Functions
Discovery Activity
Name:
1. Given f  x   2 x  6 .
(a) Graph f  x  . Is f a one-to-one function?
Why does it matter?
(b) The point (1, 8) lies on f. Write this point in
function notation: f   
(c) Find f   x  .
(d) Evaluate f   x  at the point (1, 8).
(e) Find f
(f) Is f
1
1
 x
and graph it.
a one-to-one function?
How would you know without graphing it?
(g) Find  f
1
  x  .
(h) What point on f 1 corresponds to the point (1, 8) that lies on f ? Write this point in
function notation: f 1   
(i) Evaluate  f
1
  x  at the point on
f
1
 x
that you found in (h).
(j) What do you notice about your answers to (d) and (i)?
2. Given f  x   x2 for x  0 .
(a) Graph f  x  . Is f a one-to-one function?
(b) Find f   x  .
1
(c) Find f
(d) Is f
1
 x  , restricting it if necessary, and graph it.
a one-to-one function?
(e) Find  f
1
  x  .
(f) Complete each statement:
i.
f  3  ____
iv.
1
f    ____
5
ii.
f   3  ____
v.
1
f     ____
5
iii.
f
vi.
f
1
 ____   3
  x  at the points on
vii.
Evaluate  f
viii.
Notice anything interesting about f  and  f
1
f
1
 x
1
 ____  
1
5
that you found in iii and vi.
1
 ?
3. Given f  x   3 x .
(a) Graph f  x  . Is f a one-to-one function?
(b) Find f   x  .
1
(c) Find f
(d) Is f
1
 x  , and graph it.
a one-to-one function?
(e) Find  f
1
  x  .
(f) Complete each statement:
i.
f 8  ____
iv.
 1 
f     ____
 27 
ii.
f  8  ____
v.
 1 
f      ____
 27 
iii.
f
vi.
f
1
 ____   8
  x  at the points on
vii.
Evaluate  f
viii.
Notice anything interesting about f  and  f
1
f
1
 x
1
 ____   
1
27
that you found in iii and vi.
1
 ?
4. Given f  x   x3  3x  5 .
(a) Is f a one-to-one function? How can you tell without graphing it?
(b) There is a point on f in which the y-coordinate is 9. What is the x-coordinate at this
point?
Write this point in function notation: f
(c) What point on f
1
corresponds to the point you found in (b) that lies on f ?
Write this point in function notation: f
(d) Find  f
1
 
1
 
  9 without finding a function for
f
1
. Do this below, and list the steps you
used.
__________________________________________________________________________
Can you write a rule for finding the derivative of the inverse of a function without actually
finding the inverse?
Given (a, b) is a point on y  f  x  , then (
 f  b   
1
,
) is a point on y  f 1  x  .
5. Given f  x   sin x and its graph below.
(a) Is f a one-to-one function?
  
(b) We will restrict f  x   sin x to   ,  . Is f  x  one-to-one on this domain?
 2 2
f 1  x   sin 1 x
f  x   sin x
(c) Complete each statement:
i.
f   0  ____
means that
 f   0  ____
ii.
 
f     ____
2
means that
 f   ____   ____
iii.
 
f      ____
 2
means that
 f   ____   ____
iv.
 
f     ____
4
means that
 f   ____   ____
1
1
1
1
(d) Justify your answer to iv. using derivative rules for f  x   sin x and f 1  x   sin 1 x .