The inverse of a function reverses its - coordinates. For a function , if is the input and is the corresponding output, then for the inverse is the input and would be the corresponding output.
To begin, since describes the input-output relationship of the function, replace with in the function rule.
Because the inverse of a function reverses and the variables can be switched. Notice that every other piece in the function rule remains the same.
Just as shows the input-output relationship of so does Thus, replacing with gives the rule for the inverse of
Notice that in the input is multiplied by decreased by and divided by From the rule of it can be seen that undergoes the inverse of these operation in the reverse order. Specifically, is multiplied by increased by and divided by
Some of the coordinates of the function are shown in the table. Find then graph and on the same coordinate plane.
An inverse of a function reverses its - and -coordinates. When a function is expressed as a table of values, finding its inverse means switching the coordinates. For example, the point on becomes on The following table describes
We can graph both and by marking the points from both tables on the same coordinate plane.
The graphs of a function and its inverse have a noteworthy relationship. In the coordinate plane, the function are graphed. Notice that the points of are the reversed points of
Because the coordinates of the points are reversed, is a reflection of in the line
The Vertical Line Test determines if the graph of a relation is a function, or if each -value corresponds with exactly one -value. For a function to be invertible, This leads to the Horizontal Line Test, which is performed by moving an imaginary horizontal line across the graph of a function. If this line intersects the graph more than once anywhere, the function is not invertible.