The document discusses inverse functions. It defines an inverse function as switching the x's and y's in a function's ordered pairs. If this inversion results in another function, it is called the inverse function. The domain of the original function becomes the range of the inverse function, and vice versa. It provides examples of finding the expressions for inverse functions and checking if they are correct inverses. It also discusses the differences between one-to-one and many-to-one functions, and uses the horizontal line test to determine which is being graphed. Exercises are provided to find specific inverse functions.

1. Inverse Functions

2. Inverses ...
Graphically speaking ...
A function can be inverted by switching all of
the x's with the corresponding y's in the
functions set of ordered pairs.
If this inversion is still a function, one y for
every x, it is called an
-1
inverse function f (x)
This is the same as reflecting each point on
the functions curve across the line y = x

3. If point (a, b) lies on the function f(x) then the
point (b, a) will lie on the inverse function
-1
f (x)
The domain of the function of f(x) will
then become the range of the function
-1
f (x)

4. Example
f(x) = 2x - 1
-1
f(2 ) = 3 therefore the inverse f (3) = 2
Find the expression for the inverse function
-1
f (x) and then check to see if the above
holds true
The steps
1) Test to see if the function is one to one, if
it is not the function will not have an invere
2) Replace f(x) by y
3) Interchange x with y
4) Solve for y
-1
5) replace the y with f (x)

5. Many-to-one function
More than one member of the domain is mapped to a single
member of the range.
http://fooplot.com
1 -2
mapping
4 5
9 -2
input output
range
domain

6. One-to-one function
Function where each (one) member of the domain is mapped to
a unique (one) member of the range.
http://fooplot.com
1 -2
4 5
9 7

7. Horizontal line test
Used only on the graphs of functions. (i.e the vertical line test
has already been satisfied).
Sweep a horizontal line across the graph of any function if the line
crosses the graph more than once the function is many-to-one if the
line crosses the graph everywhere exactly once then the function is
one-to-one.

8. For each function find the it's inverse
a) f(x) = 3x
b) g(x) = x - 1
c) f(x) = 2x - 1
2
d) h(x) = x
3
e) f(x) = 2x - 4 http://fooplot.com/

9. 3
e) f(x) = 2x - 4 http://fooplot.com/

10. Example
f(x) = 2x - 1
-1
f(2 ) = 3 therefore the inverse f (3) = 2
Find the expression for the inverse function
-1
f (x) and then check to see if the above
holds true
The steps
1) Test to see if the function is one to one, if
it is not the function will not have an invere
2) Replace f(x) by y
3) Interchange x with y
4) Solve for y
-1
5) replace the y with f (x)

11. Find the inverse
Example
If the function f(x) = 5x + 3
-1
find f (4)

12. Assignment
Exercise 52
1-6