The Inverse of a Function

Inverting by definition is undoing; therefore, the order of operations is used backwards in order to inverse a function.

Two functions f and g are INVERSES of each other if:

AND

It is possible for f(g(x)) = x and then g(f(x)) =not x

The inverse of a function is represented by the symbol:

The inverse of a function of a graph reflects the line: y=x

ex.

To find the inverse of a function algebraically, take a function, switch the x and y, then solve for y

ex.

f(x)= 2x+5

x= 2y+5

x-5 =2y

To inverse points on a graph, simply switch the x and y coordinates and plot.

ex.

given points: (2,5) (3,6) (4,7)

inverse points: (5,2) (6,3) (7,4)

A function is one-to-one if:

f(a)=f(b); a=b

The function has an inverse if and only if the function is one-to-one.

ex.

f(x)=2x+5

f(a)=2a+5

f(b)=2b+5

2a+5=2b+5

2a=2b

a=b

This means that for every "y" value, there is only one "x"; therefore it passes the horizontal line test.

Functions that are not one-to-one include even powered functions and absolute value functions