OnetoOne Functions Inverse Functions Finding the Inverse of

Finding the Inverse of a Function � Inverse Functions Domain and Ranges swap places.

One-to-One the inverse of a function f is itself a function, then f is

Horizontal Line Test � If every horizontal line intersects the graph of a function

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

Inverse Function Symbol � The inverse function of f is denoted by the symbol

Symmetry �A function and its inverse are symmetric with respect to the line y

Verifying Inverse Functions � Do the composition of the two functions. � If the

Graphing the Inverse Function � Swap the order of the ordered pairs. � In

Procedure for Finding the Inverse � First change f(x) to y � Swap the

Procedure for Finding the Inverse � Examples � More Examples � Book Example

Finding the Range of a Function � Remember that domain of the original function

Can Restrict Domain of an Original Function � We do this so that the

Applications � The demand for corn obeys the equation P(x) = 300 – 50

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One-to-One Functions; Inverse Functions

Finding the Inverse of a Function � Inverse Functions Domain and Ranges swap places. � Examples: 1. Given Elements 2. Given ordered pairs 3. Given a graph

One-to-One 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. � When other words for every x there is a unique y and for every y there is a unique x. � In

One-to-One Functions

Not One-to-One Functions

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

Theorem �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.

Inverse Function Symbol � 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).

Domain and Range Swap Places

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

Verifying Inverse Functions � 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.

Graphing the Inverse Function � 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.

Procedure for Finding the Inverse � 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.

Procedure for Finding the Inverse � Examples � More Examples � Book Example

Finding the Range of a Function � 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.

Can Restrict Domain of an 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)

Applications � The demand for corn obeys the equation P(x) = 300 – 50 x, 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?