The inverse of a function reverses its x-y coordinates. If, for a function f, x is an input and y is its corresponding output, for the inverse, f-1,y is the input and x would be the corresponding output. f(x)=y⇔f-1(y)=x
Some function families are inverses of each other. This is because some functions undo each other. For example, x2 and ±x are inverses because radicals and exponents (with the same index) undo each other.To begin, since f(x)=y describes the input-output relationship of the function, replace f(x) with y in the function rule. f(x)=32x−1⇔y=32x−1
Because the inverse of a function reverses x and y, the variables can be switched. Notice that every other piece in the function rule remains the same. y=32x−1⇒x=32y−1
Just as f(x)=y shows the input-output relationship of f, so does f-1(x)=y. Thus, replacing y with f-1(x) gives the rule for the inverse of f.
y=23x+1⇔f-1=23x+1
Notice that in f, the input is multiplied by 2, decreased by 1 and divided by 3. From the rule of f-1, it can be seen that x undergoes the inverse of these operation in the reverse order. Specifically, x is multiplied by 3, increased by 1, and divided by 2.
Some of the coordinates of the function g are shown in the table. Find g-1, then graph g and g-1 on the same coordinate plane.
x | g(x) |
---|---|
-4 | 3 |
-2 | 2 |
0 | 1 |
2 | 0 |
4 | -1 |
An inverse of a function reverses its x- and y-coordinates. When a function is expressed as a table of values, finding its inverse means switching the coordinates. For example, the point (-4,3) on g becomes (3,-4) on g-1. The following table describes g-1(x).
x | g-1(x) |
---|---|
3 | -4 |
2 | -2 |
1 | 0 |
0 | 2 |
-1 | 4 |
We can graph both g and g-1 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 f(x)=2x−3 and its inverse f-1(x)=2x+3 are graphed. Notice that the points of f-1 are the reversed points of f.
Because the coordinates of the points are reversed, f-1(x) is a reflection of f(x) in the line y=x.
A function is said to be invertible if its inverse is also a function. An example of an invertible function is f(x)=0.5x−2, because its inverse, f-1(x)=2x+4, is a linear function. Consider the function g(x)=x2, whose inverse is ±x.
Notice that the inverse fails the Vertical Line Test. Thus, it is not a function, which, in turn means that g is not invertible. However, if the domain of g is restricted to x≥0, the inverse, g-1(x)=x, is a function.
The Vertical Line Test determines if the graph of a relation is a function, or if each x-value corresponds with exactly one y-value. For a function to be invertible, each ymust corexactly o-valuerespond tone x-value. 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.