To find the inverse of the function $f(x) = x^3 + 2x - 1$, we follow these steps. Note that the inverse of a function $f^{-1}(y)$ undoes the action of $f(x)$. Given that $a = 2$, we also need to compute $f^{-1}(2)$ after finding the inverse.

Steps:

1. Rewrite the function:
   Start with the equation $y = f(x) = x^3 + 2x - 1$, and solve for $x$ in terms of $y$.

   $y = x^3 + 2x - 1$ Add 1 to both sides: $y + 1 = x^3 + 2x$

2. Solving for $x$:
   The equation $x^3 + 2x = y + 1$ is not easily solvable using elementary algebraic methods because it's a cubic equation. Solving for $x$ explicitly in terms of $y$ may require numerical or approximation methods, such as Newton's method, depending on the desired accuracy.

3. Find $f^{-1}(2)$:
   Given $a = 2$, we are asked to find $f^{-1}(2)$. This means we need to find $x$ such that $f(x) = 2$. Set up the equation:

   $x^3 + 2x - 1 = 2$ Simplify: $x^3 + 2x - 3 = 0$

   We will solve this cubic equation for $x$.

Solving $x^3 + 2x - 3 = 0$:

We can try possible rational roots using the Rational Root Theorem, and after some trial and error, we find:

$x = 1$

Substituting $x = 1$ into the original equation confirms this:

$1^3 + 2(1) - 1 = 2$

Thus, $f^{-1}(2) = 1$.

Summary:

• The inverse function is not easily expressible in a simple algebraic form, but we determined that $f^{-1}(2) = 1$.

Would you like a detailed explanation of the numerical methods for solving cubic equations?

Here are 5 related questions:

1. What are the general methods for finding the roots of a cubic equation?
2. How does Newton's method help in finding inverse functions numerically?
3. Can all functions have inverses? What conditions must be met?
4. How do you verify if two functions are inverses of each other?
5. How do transformations affect the graph of an inverse function?

Tip: Not all functions have easily expressible inverses. In many cases, numerical methods or approximations are needed to find inverse values.