Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function f is another function f-1 such that they undo each other.
f(f-1(x))=xandf-1(f(x))=x
Also, if x is the input of a function f and y its corresponding output, then y is the input of f-1 and x its corresponding output.
Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|
f(f-1(x))=?x | 2f-1(x)−3=?x | 2(2x+3)−3=?x | x=x ✓ |
f-1(f(x))=?x | 2f(x)+3=?x | 22x−3+3=?x | x=x ✓ |
Therefore, f and f-1 undo each other. The graphs of these functions are each other's reflection across the line y=x. This means that the points on the graph of f-1 are the reversed points on the graph of f.
LHS⋅3=RHS⋅3
LHS+1=RHS+1
LHS/2=RHS/2
Rearrange equation
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.
Because the coordinates of the points are reversed, f-1(x) is a reflection of f(x) in the line y=x.