1. One-to-One Functions (Section 3.7, pp. 280-285) and Their Inverses
Activity 29:
One-to-One Functions (Section 3.7, pp )
and Their Inverses

2. Definition of a One-One Function:
A function with domain A is called a one-to-one function if no two elements of A have the same image, that is,
f(x1) ≠ f(x2) whenever x1 ≠ x2.
An equivalent way of writing the above condition is:
If f(x1) = f(x2), then x1 = x2.

3. Horizontal Line Test:
A function is one-to-one if and only if no horizontal line intersects its graph more than once.

4. Example 1: Show that the function f(x) = 5 − 2x is one-to-one.
This line clearly passes the horizontal line test. Consequently, it is a one to one functions.

5. Example 2:
Graph the function f(x) = (x−2)2 −3. The function is not one-to-one: Why? Can you restrict its domain so that the resulting function is one-to-one?
The function is not one to one because it does not pass the horizontal line test.
Restricting the domain to all
x> 2
makes the function one to one.

6. Definition of the Inverse of a Function:
Let f be a one-to-one function with domain A and range B. Then its inverse function f−1 has domain B and range A and is defined by
f−1(y) = x if and only if f(x) = y,
for any y ∈ B.

7. Example 3: Suppose f(x) is a one-to-one function.
If f(2) = 7, f(3) = −1, f(5) = 18, f−1 (2) = 6 find:
f−1(7) =
f(6) =
f−1(−1) =
f(f−1(18)) = 2 2 3
f(5) = 18

8. If g(x) = 9 − 3x, then g−1(3) =

9. Property of Inverse Functions:
Let f(x) be a one-to-one function with domain A and range B.
The inverse function f−1(x) satisfies the following “cancellation” properties:
f−1(f(x)) = x for every x ∈ A
f(f−1(x)) = x for every x ∈ B
Conversely, any function f−1(x) satisfying the above conditions is the inverse of f(x).

10. Example 4:
Show that the functions f(x) = x5 and g(x) = x1/5 are inverses of each other.

11. Example 5:
Show that the functions
are inverses of each other.

13. How to find the Inverse of a One-to-One Function:
Write y = f(x).
Interchange x and y.
Solve this equation for x in terms of y (if possible). The resulting equation is y = f−1(x).

14. Example 6:
Find the inverse of f(x) = 4x−7.

15. Example 7:
Find the inverse of

16. Example 8:
Find the inverse of

17. Graph of the Inverse Function:
The principle of interchanging x and y to find the inverse function also gives us a method for obtaining the graph of f−1 from the graph of f. The graph of f−1 is obtained by reflecting the graph of f in the line y = x.
The picture on the right hand side shows the graphs of:

18. Example 9: Find the inverse of the function
Find the domain and range of f and f−1. Graph f and f−1 on the same Cartesian plane.

19. Domain for f(x)
Range for f(x)
Domain of f-1(x)
Range of f-1(x)