Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{3 x}{\sqrt{4 x^{2}+1}} $$

Short answer

The function \(f(x) = \frac{3x}{\sqrt{4x^2+1}}\) is a rational function with domain as all real numbers, a horizontal asymptote at y=0 and no extrema.

Step-by-step solution

Identify the Function Type

Look at the given function \(f(x) = \frac{3x}{\sqrt{4x^2+1}}\). It is a rational function, where the numerator is a first degree polynomial and the denominator is a square root of a second degree polynomial.

Determine the Domain and Asymptotes

For the domain, since the function is a fraction, it's necessary to avoid values of x that would make the denominator equal to zero. However, for the given function, the denominator \( \sqrt{4x^2+1} \) is always positive for any real number. Hence, the domain of the function is all real numbers. From the form of the function it can be seen that as x approaches both positive and negative infinity, the function approaches 0. Therefore, the function has a horizontal asymptote at y=0.

Analyze Extrema

Differentiate the function and set the derivative equal to zero to find any extrema. It's found that the function does not have any extrema as the derivative of the function is always positive.

Graph the Function

By using a computer algebra system, the shape of the function, domain, horizontal asymptote and absence of extrema can be confirmed from the graph.