1.9 | Inverse Functions


Let \(f\) and \(g\) be two functions such that
$$
f(g(x)) = x,
$$
for every \(x\) in the domain of \(g\) and
$$
g(f(x)) = x,
$$
for every \(x\) in the domain of \(f.\)
Under these conditions, the function \(g\) is called the inverse function of the function \(f.\) The function \(g\) is denoted by \(f^{-1}\) (read “f-inverse”). So,
$$
f(f^{-1}(x)) = x
$$
and
$$
f^{-1}(f(x)) = x.
$$
The domain of \(f\) must be equal to the range of \(f^{-1},\) and the range of \(f\) must be equal to the domain of \(f^{-1}.\)
Horizontal Line Test A function \(f\) has an inverse function if and only if no horizontal line intersects the graph of \(f\) at more than one point.
A function \(f\) is one-to-one if each value of the dependent variable corresponds to exactly one of the independent variable. A function \(f\) has an inverse function if and only if \(f\) is one-to-one.
Finding an Inverse Function

  1. Use the Horizontal Line Test to decide whether \(f\) has an inverse function.
  2. In the equation for \(f(x),\) replace \(f(x)\) by \(y.\)
  3. Interchange the roles of \(x\) and \(y,\) and solve for \(y.\)
  4. Replace \(y\) by \(f^{-1}(x)\) in the new equation.
  5. Verify that \(f\) and \(f^{-1}\) are inverse functions of each other by showing that the domain of \(f\) is equal to the range of \(f^{-1}\), the range of \(f\) is equal to the domain of \(f^{-1},\) and \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x. \)

E 1.9 Exercises