Problem 19 Find $f^{-1}(a)$ for the funct... [FREE SOLUTION]

Chapter 0: Problem 19

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

Expert verified

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.

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91影视

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

Step by step solution

Identify the given function and its interval

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

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

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

State the final result

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