1.5 Parametric Relations and Inverses: Relations Defined Parametrically Another way to define functions or relations is to define both elements of the ordered pair (x, y) in terms of another variable t, called a parameter. EX: Inverse Relations and Inverse Functions: An inverse relation is found by switching the domain and range of every ordered pair in the original relation. Every ordered pair (a, b) becomes (b, a). Every relation has an inverse and since every function is a relation, every function has an inverse as well. However, not every function has an “Inverse Function.” Recall that for a relation to be a function it must pass the vertical line test. When we switch the domain and range of a function the result does not always pass this test. EX: Name a common function whose inverse would not be a function? For any relation to have an inverse function it must pass the horizontal line test. EX: Which of the graphs below are graphs of: a) relations that are functions? b) relations that have inverses that are functions? •
A function whose inverse is a function has a graph that will pass both the vertical line test and the horizontal line test. This type of function is called one-­‐to-­‐one, since every x is paired with a unique y and every y with a unique x. Finding an Inverse Function Algebraically: To find an inverse function algebraically, swap the x and y variables of the original equation and solve for y. x
EX: Find the inverse function of f (x) =
x +1
Finding an Inverse Function Graphically: The graphs of the functions will be reflections of each other over the line y = x Below is an example of what this would look like: Verifying Inverse Functions: Ex: Show algebraically that f (x) = x 3 + 1 and g(x) = 3 x − 1 are inverse functions.