Inverse Functions

Inverse Functions Domain and Ranges swap places.

Examples: 1. Given Elements

2. Given ordered pairs

3. Given a graph

When the inverse of a function f is itself a function, then f is said to be a one-to-one function. That is f is one-to-one if, for any choice of elements in the domain of f, the corresponding values in the range are unequal.

In other words for every x there is a unique y and for every y there is a unique x.

If every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one.

A function that is increasing over its domain is a one-to-one function. A function that is decreasing over its domain is a one-to-one function.

The inverse function of f is denoted by the symbol f-1

Be careful! This symbol does not mean the reciprocal of f or 1/f(x).

1 1Domain of Range of Range of Domain of f f f f

A function and its inverse are symmetric with respect to the line y = x.

Do the composition of the two functions. If the answer is x, the functions are inverses

of each other. If not, they are not inverses of each other. Be sure the functions are one-to-one first.

1 1( ( )) and ( ( ))f f x x f f x x

Swap the order of the ordered pairs. In other words, make the x the y value and

the y the x value Plot these points.

First change f(x) to y Swap the x’s and y’s Solve the equation for y Put the symbol for inverse in for y To make sure your answer is correct, do the

composition and see if you get x.

Examples More Examples Book Example

Remember that domain of the original function is the range of the inverse function and vice versa.

Find the domain of the inverse function in order to find the range of the original function.

We do this so that the inverse can now be a function.

The quadratic function can have its domain restricted to either x > 0 or x < 0 and its inverse is now a function. (Look at the horizontal line test)

The demand for corn obeys the equation P(x) = 300 – 50x, where p is the price per

bushel (in dollars) and x is the number of bushels produced, in millions. Express the production amount x as a function of the price p.

Why would this be important for a producer to know?