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

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.

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

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.

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.

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.

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

If g(x) = 9 − 3x, then g−1(3) =

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).

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

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

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).

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

Example 7: Find the inverse of

Example 8: Find the inverse of

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:

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.

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