3.6 – Inverse Functions Exchanging input and output values of a function or relation is called an inverse relation. If the inverse is a function, then we have an inverse function. The key idea of inverses is that for a point x , y on a graph, the point y , x is on its inverse graph. EXAMPLE 1: Sketch the graph of the quadratic function f which passes through the following points: (6,5) (0,5)
(5,0)
(1,0)
(4, 3)
(2, 3)
(3, 4) Graph the inverse on the same set of axes and describe the relationship between the function and its inverse.
Let f be a function. If a, b is a point on f, then b, a is a point on the graph of its inverse. The graph of the inverse of f is a reflection of the graph of f across the line y x . To find an inverse from an equation, write the function f (x) in terms of x and y, switch the x and y, and then solve for y. Call the result g(x) . EXAMPLE 2: Find g(x) , the inverse of f (x) 2x 1 .
EXAMPLE 3: Find g(x) , the inverse of f (x) 4 x 2 2x .
To determine whether an inverse of a function f is also a function, every output of f must correspond to exactly one input, making f a one-to-one function. A function f is one-to-one if f (a) f (b) implies that a b . If a function is one-to-one, then its inverse is also a function. A function is one-to-one if it passes the horizontal line test. (NOTE: a relation is a function if it passes the vertical line test and if it passes the horizontal line test, its inverse is also a function!!)