1. The problem is to plot the graph of the function $f(x) = x - \ln x$. 2. This function is defined for $x > 0$ because the natural logarithm $\ln x$ is only defined for positive $x$. 3. To understand the shape of the graph, let's analyze the function: - The function is $f(x) = x - \ln x$. - The derivative is $f'(x) = 1 - \frac{1}{x}$. 4. Find critical points by setting the derivative to zero: $$ 1 - \frac{1}{x} = 0 \\ 1 = \frac{1}{x} \\ x = 1 $$ 5. Determine the nature of the critical point by the second derivative: $$ f''(x) = \frac{1}{x^2} > 0 \text{ for } x > 0 $$ Since $f''(1) > 0$, the function has a local minimum at $x=1$. 6. Evaluate the function at $x=1$: $$ f(1) = 1 - \ln 1 = 1 - 0 = 1 $$ 7. Behavior as $x \to 0^+$: $$ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x - \ln x) = +\infty $$ because $\ln x \to -\infty$. 8. Behavior as $x \to +\infty$: $$ \lim_{x \to +\infty} f(x) = +\infty $$ since $x$ dominates $\ln x$. 9. Summary: - Domain: $x > 0$ - Local minimum at $(1,1)$ - $f(x) \to +\infty$ as $x \to 0^+$ and as $x \to +\infty$ This information helps to sketch the graph.