Pc Inverse Functions & 1 To 1

One-to-one and Inverse Functions

A function y = f (x) with domain D is one-to-one on D if and only if for every x1 and x2 in D, f (x1) = f (x2) implies that x1 = x2. A function is a mapping from its domain to its range so that each element, x, of the domain is mapped to one, and only one, element, f (x), of the range. A function is one-to-one if each element f (x) of the range is mapped from one, and only one, element, x, of the domain.

Horizontal Line Test A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) in more than one point. Example: The function y = x2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7).

y (4, 7)

(0, 7)

y=7 2 2

x

Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. b) y = x3 + 3x2 – x – 1

a) y = x3

y

-4

y

8

8

4

4 4

one-to-one

x

-4

4

not one-to-one

x

Every function y = f (x) has an inverse relation x = f (y). Function y = |x| + 1 y x 2 3 1 2 0 1 -1 -2 Domain Range

Inverse relation x = |y| + 1 y x 2 3 1 2 0 1 -1 -2 Range Domain

The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}. x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}. The inverse relation is not a function. It pairs 2 to both -1 and +1.

The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain of the inverse relation is the range of the original function. The range of the inverse relation is the domain of the original function.

The graphs of a relation and its inverse are reflections in the line y = x. Example: Find the graph of the inverse relation 3 ( x − 2) geometrically from the graph of f (x) = 4 y

The ordered pairs of f are 3 ( x − 2) given by the equation y = . 4

y=x

2

The ordered pairs of the inverse are 3 ( y − 2) given by x = . 4

-2

( y − 2) x= 4 3

2 -2

( x 3 − 2) y= 4

x

To find the inverse of a relation algebraically, interchange x and y and solve for y. Example: Find the inverse relation algebraically for the function f (x) = 3x + 2. y = 3x + 2

Original equation defining f

x = 3y + 2

Switch x and y.

3y + 2 = x

Reverse sides of the equation.

( x − 2) y= 3

Solve for y.

To calculate a value for the inverse of f, subtract 2, then divide by 3.

For a function y = f (x), the inverse relation of f is a function if and only if f is one-to-one. For a function y = f (x), the inverse relation of f is a function if and only if the graph of f passes the horizontal line test. If f is one-to-one, the inverse relation of f is a function called the inverse function of f. The inverse function of y = f (x) is written y = f -1(x).

Example: From the graph of the function y = f (x), determine if the inverse relation is a function and, if it is, sketch its graph. The graph of f passes the horizontal line test.

y y = f -1(x) y=x y = f(x) x

The inverse relation is a function. Reflect the graph of f in the line y = x to produce the graph of f -1.

The inverse function is an “inverse” with respect to the operation of composition of functions. The inverse function “undoes” the function, that is, f -1( f (x)) = x. The function is the inverse of its inverse function, that is, f ( f -1(x)) = x. Example: The inverse of f (x) = x3 is f -1(x) = 3

3

f ( f(x)) = x = x and f ( f -1(x)) = ( 3 x )3 = x. -1

3

x.

x +1 Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1. 2 ( f ( x) + 1) ((2 x − 1) + 1) 2x g( f(x)) = = = =x 2 2 2 x +1 f(g(x)) = 2g(x) – 1 = 2( ) – 1 = (x + 1) – 1 = x 2

It follows that g = f -1.