1. Inverse Functions

2.  Inverse Functions Domain and Ranges swap places.  Examples: 1. Given ElementsGiven Elements 2. Given ordered pairs 3. Given a graphGiven a graph

3.  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.

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

7.  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.

8.  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).

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

11.  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.

12.  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.

13.  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.

14.  Examples Examples  More Examples More Examples  Book Example

15.  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.

16.  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)

17.  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?