f^-1f and f f^-1

Composite function f^-1f

The composite function $f^{-1} f$ is defined as: $$ \boxed{ f^{-1} f(x) = x } $$

The domain and range are $$ \boxed{ D_{f^{-1} f } = D_f }$$ $$ \boxed{ R_{f^{-1} f } = D_f } $$

Example

Given $$ g : x \mapsto e^x, \phantom{000} x \in \mathbb{R}, 0 < x \le 1 $$

and $$ D_g = (0, 1] \text{ and } R_g = (1, e] $$

The composite function $g^{-1} g$ is defined as $$ g^{-1} g (x) = x $$ $$ D_{g^{-1}g} = (0, 1] $$ $$ R_{g^{-1}g} = (0, 1] $$

The graph of $y = g^{-1} g$ is:

Composite function f f^-1

The composite function $f f^{-1}$ is defined as: $$ \boxed{ f f^{-1}(x) = x } $$

The domain and range are $$ \boxed{ D_{f f^{-1} } = D_{f^{-1}} }$$ $$ \boxed{ R_{f f^{-1} } = D_{f^{-1}} } $$

Example

Given $$ g : x \mapsto e^x, \phantom{000} x \in \mathbb{R}, 0 < x \le 1 $$

and $$ D_g = (0, 1] \text{ and } R_g = (1, e] $$

The composite function $g g^{-1}$ is defined as $$ g g^{-1} (x) = x $$ $$ D_{g g^{-1}} = D_{g^{-1}} = R_g = (1, e] $$ $$ R_{g g^{-1}} = D_{g^{-1}} = R_g = (1, e] $$

The graph of $y = g g^{-1}$ is: