Inverse Functions

If f and g are functions such that

  1. for every x in the domain of g, f(g(x)) = x and
  2. for every x in the domain of f, g(f(x)) = x,

then we say that g is the inverse of f and write g = f–1, which is read “f inverse.” It is also true that f is the inverse of g: f = g–1.

Key Fact N7

If for some function f, f1 exists, then:

f(f-1x)=x and f-1fx=x

The inverse, f–1, of a function, f, undoes what f does. In Example 15, f multiplies a number by 3 and then subtracts 2 from it; g, which is f–1, adds 2 to a number and then divides the result by 3.

Function (f) means that x multiplies by 3 and subtracts 2, or 3x - 2. Function f^-1, is the opposite, which is x divides by 3 and adds 2. Then (f^-1  x  f)(10) = 10. Input of 10 into function f is 28, and input of 10 into function f^-1 is 10. Finally, (f x f^-1)(10) = 10. Input of 10 into function f^-1 is 4, and input of 10 into function f is 10.

Not every function has an inverse, but many do. On the Math 1 test, you may be asked to find the inverse of a particular function. The procedure to do this is given in KEY FACT N8.

Key Fact N8

If f is a function of x, to find f1, first write y = f(x). Then interchange x and y and solve for y.

So, f-1x=x+23 . Note that f–1, which is function g in Example 15, simply undoes what f does: f multiplies a number by 3 and then subtracts 2; f–1 adds 2 to a number and then divides the result by 3.