/31.6__Inverse_Functions

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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!!)


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