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 \(f^{-1}.\)
$$
f(g(x)) = x,
$$
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
- Use the Horizontal Line Test to decide whether \(f\) has an inverse function.
- In the equation for \(f(x),\) replace \(f(x)\) by \(y.\)
- Interchange the roles of \(x\) and \(y,\) and solve for \(y.\)
- Replace \(y\) by \(f^{-1}(x)\) in the new equation.
- 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 \(f^{-1}(f(x)) = x. \)