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!!)

EXAMPLE 4: Graph each function below and determine whether the function is one-to-one. If so, graph its inverse function: a) f (x)  3x 5  2x 4  3x 3  x  2

 Note that f is always ____________________, so it passes the ___________________________and is _____________________; its inverse is a ________________ that is also always ______________________. b) g(x)   0.5x 3  x  1

 Note that g is ____________________ AND ____________________, so it does NOT pass the ________________________ and it is NOT _____________________ and its inverse is NOT a function!! c) h(x)   2  0.8 x 3

 Note that h is always ____________________, so it passes the ________________________ and is _____________________ and its inverse is also always ______________________. Let f be a function. TFAE (the following are equivalent):  The inverse of f is a function  f is one-to-one  the graph of f passes the Horizontal Line Test The inverse function, if it exists, is written as f  1 where:  if y  f (x) , then x  f  1 (y)  the notation f  1 does not mean

1 f

 When a function is not one-to-one, it is possible to create an inverse function by restricting the domain of the function to the part that that is one-to-one. EXAMPLE 5: Find an interval on which the function f (x)  x 2  3 is one-to-one, and find f  1 on that interval.

The inverse of a function f sends each output of f back to the input it came from, that is: f  a   b exactly when f  1  b   a (a is sent to b under f exactly when b gets sent back to a under f  1 )

A one-to-one function f and its inverse function f  1 have these properties:

f

f

1

f



 x 

 f  1  f  x    x for every x in the domain of f



f 1 x  f f

1

 x 

 x for every x in the domain of f  1

Also, any two functions having both properties are one-to-one and are inverses of each other. IN OTHER WORDS: If you compose a function with its inverse, you will get x.

EXAMPLE 6: Assume f is a one-to-one function. a) If f  3  10, find f  1 10 

b) If f  1  7   3, find f   3

2x 3x . Use composition to verify that f and g are inverses of each and g(x)  3 x x 2 other. (In other words, compute f g and g f and show that the result is x.)

EXAMPLE 7: Let f (x) 