Ch 9 – Properties and Attributes of Functions 9.5 – Functions and their Inverses

What is the inverse of a function?  The inverse of a function f(x) “undoes” f(x)  Its graph is a reflection of f(x) across the line y = x  Sometimes, the inverse ends up NOT being a function.  If the inverse is a function, then it is denoted as f -1 (x)

Horizontal-Line Test If any horizontal line passes through more than one point on the graph of a relation, the inverse relation is NOT a function. The inverse relation is a function The inverse relation is not a function

Use the horizontal-line test to determine whether the inverse of each relation is a function.

Finding Inverses of functions  To find the inverse of a function, switch the x and y, then solve for y.  Example: Find the inverse of f(x) = x 2

Find the inverse f -1 (x) of Determine whether it is a function, and state its domain and range.

When both a relation and its inverse are functions, the relation is called a one-to- one function.  In a one-to-one function, each y-value is paired with EXACTLY one x-value.  You can use composition of functions to verify that two functions are inverses.  If f(g(x)) = g(f(x)) = x then f(x) and g(x) are inverse functions.

Determine by composition whether each pair of functions are inverses.