The problem given is about the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by:

$f(x) = e^x + x - 1$

We are asked to solve the following parts:

a) Show that $f$ is bijective (i.e., one-to-one and onto).

b) Find the inverse function $f^{-1}$.

c) Find $f(0)$.

d) Solve the equation $e^x + x = 3x^2$.

e) Find the common points of the curves $C_f$ and $C_{f^{-1}}$.

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Solution:

a) Show that $f$ is bijective

• Injectivity (One-to-One):
  We need to show that if $f(x_1) = f(x_2)$, then $x_1 = x_2$. Let's consider: $f(x_1) = f(x_2) \implies e^{x_1} + x_1 - 1 = e^{x_2} + x_2 - 1$ $e^{x_1} - e^{x_2} = x_2 - x_1$ Since $e^x$ is strictly increasing, $e^{x_1} - e^{x_2} > 0$ if $x_1 > x_2$, which implies $x_2 - x_1 > 0$, a contradiction. Thus, $x_1 = x_2$, proving injectivity.

• Surjectivity (Onto):
  We must show that for every $y \in \mathbb{R}$, there exists an $x \in \mathbb{R}$ such that $f(x) = y$. Given the nature of the function (sum of $e^x$ and a linear term), as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$, and as $x \rightarrow \infty$, $f(x) \rightarrow \infty$. By the Intermediate Value Theorem, $f(x)$ takes every real value. Therefore, $f(x)$ is surjective.

Since $f(x)$ is both injective and surjective, it is bijective.

b) Find the inverse function $f^{-1}$

To find the inverse function $f^{-1}(y)$, solve the equation:

$y = e^x + x - 1$

This equation is generally transcendental, meaning it cannot be solved algebraically for $x$. However, we can express the inverse implicitly as the function $f^{-1}(y)$ is the solution $x$ to the equation:

$x = W(e^y) - y$

where $W$ is the Lambert W function, which is used for such transcendental equations.

c) Find $f(0)$

$f(0) = e^0 + 0 - 1 = 1 - 1 = 0$

So, $f(0) = 0$.

d) Solve the equation $e^x + x = 3x^2$

This equation does not have a simple algebraic solution. Numerical methods (like Newton's method) would generally be used to find approximate solutions. However, by substitution or graphical methods, one might be able to find solutions or verify that no real solutions exist.

e) Find the common points of $C_f$ and $C_{f^{-1}}$

The common points of $C_f$ and $C_{f^{-1}}$ satisfy:

$f(x) = x$

So, solve the equation:

$e^x + x - 1 = x$ $e^x = 1$

This implies $x = 0$. Therefore, the only common point is $(0, 0)$.

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Further questions:

1. How would you find the derivative of the function $f(x)$ and what does it tell you about the behavior of $f(x)$?
2. Can you analyze the concavity of the function $f(x)$?
3. What are the possible methods to solve a transcendental equation like $e^x + x = 3x^2$ numerically?
4. How does the Lambert W function work and where is it applied?
5. Can the results of this function be applied to any real-world situation, such as in physics or economics?

Tip: Understanding bijective functions is crucial in many fields of mathematics, as it guarantees that every element in the domain has a unique counterpart in the range and vice versa.