Example 6: The inverse of the exponential function - the natural logarithm

The parametric curve or trajectory for these parametric equations will be the inverse of $y=e^x$ , or $y=\ln x$ .

To obtain a parametric curve matching the graph of the inverse of a function, let $f$ in the definition $x=f(t)$ be the function we want to plot the inverse of, and let $g$ in $y=g(t)$ be the identity function, that is, let $y=t$ .

$$\left. \begin{array}{lclcl} x &=& f(t) &=& e^t\\ y &=& g(t) &=& t\\ \end{array}\right\}\quad -2\leq t\leq 2$$

Any time $x=f(t)$ is some function of $t$ and $y=t$ is the identity function,

the parametric curve will be the same as the graph of $y=g(x)=f^{-1}(x)$ .

The above graph matches that of $y=\ln x$ , the inverse of the function $y=e^x$ .