AP CALCULUS-AB
Section Number:
LECTURE NOTES
Topics: Inverse Functions
MR. RECORD
Day: 4
5.3
Inverse Functions
Recall: From College Algebra, a function can be represented by a set of ordered pairs. For instance, the function
f (x)  x  3 and A  1,2,3,4 to B  4,5,6,7 can be written as f : (1,4),(2,5),(3,6),(4,7).
By interchanging the first and second coordinates of each ordered pair, you can form the inverse function of f. This
function is denoted by f 1 . It is a function from B to A, and can be written as f 1 : (4,1),(5,2),(6,3),(7,4).
Definition of Inverse Function
A function g is the inverse function of the function f if
f (g(x))  x for each x in the domain of g and g( f (x))  x for each x in the domain of f.
The function g is denoted by f 1 and is read “f inverse”
Thus, the domain of f is equal to the range of f 1 and vice versa. Additionally it is true that
f ( f 1 (x))  x and f 1 ( f (x))  x as a function and its inverse tend to “undo” each other.
Example 1:
a. Show that the functions are inverse functions of each other.
x 1
2
f (x)  2x 3  1 and g(x)  3
b. Sketch the graphs of both functions on the same set of axes.
y
2
1
x
-3
-2
-1
1
-1
-2
-3
2
3
Reflective Property of Inverse Functions Theorem
The graph of f contains the point (a,b) if and only if the graph of f 1 contains the point (b,a)
TI-Nspire Activity: Inverse Functions on the TI-Nspire
Create a New Document Calculator page, and enter
menu.
x → f ( x ) You can access the → from the
Next, enter the following Solve( f (y)  x,y ).
What did the calculator return as the result?
Next enter x 2 → g(x)
Insert a Graph page, and enter f ( x ) and g(x) as f 1( x) and f 2( x) respectively.
What do you notice about the two graphs?
two different ways.
Method 1: of
On
graph page,
enter f1(x) as
Existence
ana Inverse
Function
f1(x) = zeros(x2+4y2-4,y)
Method 2: On a graph page, press b ,
The Existence of an Inverse Function Theorem
3: Graph/Entry
Edit, 2: Equation, 6: Conic
1. A function has an inverse if and
only if it is one-to-one
and
enter
the
coefficients
in
the
correct
2. If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse
function. boxes. Be sure set your relation equal to
zero first.
Example 2: Which of the following has an inverse function
a. y  x 3  x  1
b. y  x 3  x  1
Guidelines for Finding an Inverse Function
1. Use the Inverse Function Existence Theorem above to determine whether the function given by y=f(x) has an inverse function.
2. Solve for x as a function of y: x  g(y)  f 1 (y) .
3. Interchange x and y. The resulting equation is y  f 1 (x) .
4. Define the domain of f 1 to be the range of f .
5. Verify that f ( f 1 (x))  x and f 1 ( f (x))  x.
Derivative of an Inverse Function
Theorem for Continuity and Differentiability of Inverse Function
Let f be a function whose domain is an interval I. If f has an inverse function, then the
following statements are true.
1. If f is continuous on its domain, then f 1 is continuous on its domain.
2. If f is increasing on its domain, then f 1 is increasing on its domain.
3. If f is decreasing on its domain, then f 1 is decreasing on its domain.
4. If f is differentiable at c and f (c)  0 , then f 1 is differentiable at c.
Evaluating the Derivative of an Inverse Function
This works when it is easy to generate the inverse function.
a) Find the inverse function by interchanging x and y and solving for y.
b) Take the derivative of this new y. That will be the derivative of the inverse
function.
c) Plug in your given k value (which is some vallue for x).
Example 3:
If f (x)  x 2 , x  0 , find the derivative of f 1 (x) at x  4.
This method is used when finding the inverse of the function is difficult or impossible.
a) Find the inverse function by interchanging x and y. (Don’t solve for y)
dy
b) Find
implicitly
dx
dy
c) Solve for
. It will be in terms of y.
dx
d) Replace the value of k for x in your inverse function from Step ‘a’ above and solve
for y.
dy
e) Plug that value of y into
.
dx
Example 4:
Find the derivative of the inverse function of f (x)  x 3 - 4 x 2  7 x -1 at x  1 .
Example 5:
Find the derivative of the inverse function of f (x)  e x +ln x at x  3.
It’s possible that Method 2 can be streamlined a bit by using the following formula.
The Derivative of an Inverse Function
Let f be a function that is differentiable on an interval I. If f has an inverse function f 1 ,
then f 1 is differentiable at any x for which f ( f 1 (x))  0 . Moreover,
 f  (x)  f ( f 1 (x))
1
1
Example 6:
Find the derivative of the inverse function of f (x)  x+sin x at x   using the formula above.
Example 7:
1 3
Find the derivative of the inverse function of f (x)  x +x  1 at x  3 using the formula
4
above.