Question

Find \(f^{-1}(a)\) for the function \(f\) and the real number \(a\). $$ f(x)=\frac{3}{\pi} x+\sin x ; \quad-\frac{\pi}{2}

Short answer

The inverse function \(f^{-1}(a)\) evaluated at \(a=1\) is approximately \(0.4502\). Keep in mind that this is an approximate value as we used a numerical method to find the solution.

Step-by-step solution

Identify the given function and its interval

We are given the function \(f(x) = \frac{3}{\pi}x + \sin x\) with an interval \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). We must find the inverse function and evaluate it at \(a=1\).

Express \(x\) in terms of \(y\)

To find the inverse function, we must express \(x\) in terms of \(y\). Let \(y = f^{-1}(x)\), so we have \(x = f(y) = \frac{3}{\pi}y + \sin y\). We need to solve for \(y\) in terms of \(x\). However, since this equation consists of a linear term (\(\frac{3}{\pi}y\)) and a trigonometric term (\(\sin y\)), it is not solvable algebraically. Hence, we'll need to use a numerical method to find the inverse function evaluated at a specific value of \(a\).

Set up an equation for \(f^{-1}(a)\)

We want to find \(f^{-1}(a)\) for \(a=1\). From the relationship we found earlier, this corresponds to solving the equation: $$ 1 = \frac{3}{\pi}y + \sin y $$ for \(y\).

Use a numerical method to solve for \(y\)

Since we cannot solve this equation algebraically, we'll use an iterative numerical method such as the Newton-Raphson method or the bisection method. You can use a graphing calculator, computer software, or an online tool to find the solution for \(y\). Using a numerical method, we find that \(f^{-1}(1) \approx 0.4502\).

State the final result

The inverse function \(f^{-1}(a)\) evaluated at \(a=1\) is approximately \(0.4502\). Keep in mind that this is an approximate value as we used a numerical method to find the solution.