Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 3x/(x² + 3)

Verified step by step guidance

1. Identify the domain of the function \( f(x) = \frac{3x}{x^2 + 3} \). Since the denominator \( x^2 + 3 \) is always positive for all real numbers, the domain is all real numbers.
2. Find the intercepts: For the y-intercept, set \( x = 0 \) and solve \( f(0) = \frac{3(0)}{0^2 + 3} = 0 \). For the x-intercept, set \( f(x) = 0 \), which implies \( 3x = 0 \), so \( x = 0 \). Thus, the intercept is at the origin (0,0).
3. Determine the asymptotes: Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at \( y = 0 \). There are no vertical asymptotes because the denominator \( x^2 + 3 \) never equals zero.
4. Analyze the first derivative \( f'(x) \) to find critical points and determine intervals of increase and decrease. Use the quotient rule: \( f'(x) = \frac{(x^2 + 3)(3) - 3x(2x)}{(x^2 + 3)^2} \). Simplify and solve \( f'(x) = 0 \) to find critical points.
5. Analyze the second derivative \( f''(x) \) to determine concavity and points of inflection. Use the quotient rule again on \( f'(x) \) to find \( f''(x) \), and solve \( f''(x) = 0 \) to find points of inflection. Use this information to sketch the graph, and verify with a graphing utility.

Key Concepts

Function Analysis

Function analysis involves examining the properties of a function, such as its domain, range, intercepts, and asymptotes. For the function ƒ(x) = 3x/(x² + 3), understanding where the function is defined and how it behaves at critical points is essential for graphing it accurately.

Graphing Techniques

Graphing techniques include methods for plotting functions accurately, such as identifying key points, using symmetry, and recognizing asymptotic behavior. For ƒ(x) = 3x/(x² + 3), determining the x-intercept and analyzing the end behavior will help create a complete graph.

Use of Graphing Utilities

Graphing utilities are software tools or calculators that assist in visualizing functions. They can provide a quick way to verify the accuracy of a hand-drawn graph by showing the function's behavior over its domain, which is particularly useful for complex functions like ƒ(x) = 3x/(x² + 3).