The task is to analyze the function $f(x) = x - \ln(x+1)$ and construct its graph. Let's break this down step by step:

Step 1: Domain of the function

The logarithmic term $\ln(x+1)$ is defined when $x+1 > 0$, so: $x > -1$ Thus, the domain of the function is $x \in (-1, +\infty)$.

Step 2: First derivative for monotonicity and critical points

The derivative of $f(x)$ is: $f'(x) = 1 - \frac{1}{x+1}$ Simplify: $f'(x) = \frac{x}{x+1}$

• $f'(x) = 0$ when $x = 0$. This is a critical point.
• Analyze the sign of $f'(x)$:
  • For $x \in (-1, 0)$, $f'(x) < 0$ (function is decreasing).
  • For $x \in (0, +\infty)$, $f'(x) > 0$ (function is increasing).

Thus, $x = 0$ is a local minimum.

Step 3: Second derivative for concavity

The second derivative is: $f''(x) = \frac{d}{dx} \left( \frac{x}{x+1} \right)$ Using the quotient rule: $f''(x) = \frac{(1)(x+1) - x(1)}{(x+1)^2} = \frac{1}{(x+1)^2}$

• $f''(x) > 0$ for all $x > -1$, meaning the function is concave up everywhere in its domain.

Step 4: Behavior at boundaries

1. As $x \to -1^+$, $\ln(x+1) \to -\infty$, so $f(x) \to +\infty$.
2. As $x \to +\infty$, $\ln(x+1)$ grows slower than $x$, so $f(x) \to +\infty$.

Step 5: Key points

Evaluate the function at some points:

• At $x = 0$: $f(0) = 0 - \ln(1) = 0$.
• At $x = 1$: $f(1) = 1 - \ln(2) \approx 1 - 0.693 = 0.307$.
• At $x = 2$: $f(2) = 2 - \ln(3) \approx 2 - 1.099 = 0.901$.

Step 6: Graph sketch

1. The function decreases for $x \in (-1, 0)$ and has a local minimum at $x = 0$.
2. It increases for $x \in (0, +\infty)$.
3. The graph is concave up everywhere.
4. Asymptotic behavior near $x = -1$ and as $x \to +\infty$.

Would you like me to generate the graph for this function? Let me know if you’d like any step explained further or if you have specific questions!

Related Questions:

1. What is the role of concavity in determining the shape of the graph?
2. Can we determine the asymptotes of $f(x)$ more formally?
3. How do logarithmic functions influence domain restrictions?
4. What happens if we shift the function by adding a constant to $f(x)$?
5. How would you find the inverse of $f(x)$ if it exists?

Tip: Always check the domain of a logarithmic function before proceeding with analysis!