The inverse function (g) of some function (f(x) ) takes the value of (f(x) ) back...

 

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The inverse function (g) of some function (f(x) \) takes the value of \(f(x) \) back to \(x) again: \(g(f(x))=x \). The common technique to compute inverse functions is to set \(y=f(x) \) and solve with respect to \(x \). The formula on the right-hand side is then the desired inverse function (g(y) \). We can formulate a general procedure for computing inverse functions from an ODE problem. If we differentiate (y-f(x)) with respect to (y), we get (1= f(x) frac {dx} {dy}) by the chain rule. The inverse function we seek is (x(y)), but this function then fulfills the ODE SS \begin{equation) x'(y) = \frac{1} {f(x)} \tp \tag (68} \end {equation} $$ That (y) is the independent coordinate and \(x \) a function of (y) can be a somewhat confusing notation, so we might introduce \(u) for (x) and (t) for (y): $$ \begin{equation*} u^{\prime} (t) = \frac{1} {f(u)} \tp\end{equation*} $$ The initial condition is \(x(0)=x_r) where \(x_r) solves the equation (f(x_r)=0)) (( x(0) ) implies (y=0) and then from \(y-f(x) \) it follows that (f(x(0))=0 \)). Make a program that can use the described method to compute the inverse function of \(f(x) ), given \(x_r). Use any numerical method of your choice for solving the ODE problem. Verify the implementation for \( f(x)=2x\). Apply the method for \( f(x)=\sqrt {x} \) and plot ( f(x) \) together with its inverse function. Filename: inverse_ODE. The inverse function (g) of some function (f(x) \) takes the value of \(f(x) \) back to \(x) again: \(g(f(x))=x \). The common technique to compute inverse functions is to set \(y=f(x) \) and solve with respect to \(x \). The formula on the right-hand side is then the desired inverse function (g(y) \). We can formulate a general procedure for computing inverse functions from an ODE problem. If we differentiate (y-f(x)) with respect to (y), we get (1= f(x) frac {dx} {dy}) by the chain rule. The inverse function we seek is (x(y)), but this function then fulfills the ODE SS \begin{equation) x'(y) = \frac{1} {f(x)} \tp \tag (68} \end {equation} $$ That (y) is the independent coordinate and \(x \) a function of (y) can be a somewhat confusing notation, so we might introduce \(u) for (x) and (t) for (y): $$ \begin{equation*} u^{\prime} (t) = \frac{1} {f(u)} \tp\end{equation*} $$ The initial condition is \(x(0)=x_r) where \(x_r) solves the equation (f(x_r)=0)) (( x(0) ) implies (y=0) and then from \(y-f(x) \) it follows that (f(x(0))=0 \)). Make a program that can use the described method to compute the inverse function of \(f(x) ), given \(x_r). Use any numerical method of your choice for solving the ODE problem. Verify the implementation for \( f(x)=2x\). Apply the method for \( f(x)=\sqrt {x} \) and plot ( f(x) \) together with its inverse function. Filename: inverse_ODE.

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