Question

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)

Accepted Answer

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. 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). 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. 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. 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.

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)

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Concetti chiave

Function Analysis

Graphing Techniques

Use of Graphing Utilities