Consider the following. \[ f(x)=\ln (x)+5 \] (a) What is the domain of \( f \) ? (Enter your answer using interval notation.) \( (0, \infty) \) What is the range of \( f \) ? (Enter your answer using interval notation.) (b) What is the \( x \)-intercept of the graph of \( f \) ? (- \( x=0.0067 \) (c) Sketch the graph of \( f \).

We are given the function f(x) = ln(x) + 5. (a) Domain and Range • Domain: The natural logarithm ln(x) is defined only for x > 0. Hence, the domain is (0, ∞). • Range: The function ln(x) has a range of (–∞, ∞) and adding 5 shifts the entire graph vertically by 5. Thus, the range remains (–∞, ∞). (b) x-intercept To find the x-intercept, set f(x) = 0:   ln(x) + 5 = 0  →  ln(x) = –5  →  x = e^(–5). Using a calculator, e^(–5) ≈ 0.0067, so the x-intercept is approximately (0.0067, 0). (c) Sketching the Graph To sketch the graph of f(x) = ln(x) + 5, note the following: 1. Basic Logarithm: The graph of y = ln(x) is defined for x > 0, passes through (1, 0), and has a vertical asymptote at x = 0. It increases slowly for x > 1. 2. Vertical Shift: The +5 moves every point on y = ln(x) up by 5. Therefore, the graph now passes through (1, 5) instead of (1, 0). 3. x-intercept: As calculated, the graph crosses the x-axis at (e^(–5), 0) ≈ (0.0067, 0). 4. y-intercept: There is no y-intercept because the function is not defined at x = 0. 5. Asymptote: The vertical asymptote remains x = 0. Steps to sketch:  • Draw a vertical dashed line at x = 0 for the asymptote.  • Plot the point (e^(–5) ≈ 0.0067, 0) for the x-intercept.  • Plot the point (1, 5).  • Sketch a smooth curve that increases slowly to the right, passes through (1, 5), and continues to rise gradually.  • As x approaches 0 (from the right), note that f(x) goes to –∞ (the graph dives downward near the asymptote).  • As x increases, the graph continues to rise, but at a decreasing rate compared to linear functions. The final answers: Domain: (0, ∞) Range: (–∞, ∞) x-intercept: (e^(–5), 0) or approximately (0.0067, 0) This completes the analysis and sketch of the function f(x) = ln(x) + 5.