Verified Answer :
- Evaluate the function at \(x=9\): \(f(9) = \frac{477}{716}\).
- Find the x-intercepts by solving \(6x^2 - 9 = 0\), yielding \(\left(\pm\frac{\sqrt{6}}{2}, 0\right)\).
- Determine the y-intercept by calculating \(f(0) = \frac{9}{13}\).
- Identify the vertical asymptote at \(x = \sqrt[3]{13}\) and the horizontal asymptote at \(y = 0\).
- Final results:
- \(f(9) = \boxed{\frac{477}{716}}\)
- \(x\)-intercepts: \(\boxed{\left(\frac{\sqrt{6}}{2}, 0\right), \left(-\frac{\sqrt{6}}{2}, 0\right)}\)
- \(y\)-intercept: \(\boxed{\left(0, \frac{9}{13}\right)}\)
- Vertical Asymptote: \(\boxed{x = \sqrt[3]{13}}\)
- Horizontal Asymptote: \(\boxed{y = 0}\)
Explanation
- Problem Analysis
To analyze the function \(f(x) = \frac{6x^2 - 9}{x^3 - 13}\), we examine its behavior, intercepts, and asymptotic properties. The numerator is \(6x^2 - 9\) and the denominator is \(x^3 - 13\). The function is defined for all real numbers except where the denominator is zero, i.e., \(x^3 = 13\), which implies \(x = \sqrt[3]{13}\).
- Evaluating \(f(9)\)
To evaluate \(f(9)\), we substitute \(x = 9\) into the function expression:
\[f(9) = \frac{6(9)^2 - 9}{9^3 - 13} = \frac{6(81) - 9}{729 - 13} = \frac{486 - 9}{716} = \frac{477}{716}\]
Thus, \(f(9) = \frac{477}{716}\).
- Finding the x-intercepts
The x-intercepts occur where the function equals zero, which requires the numerator to be zero while the denominator is non-zero:
\[6x^2 - 9 = 0 \implies 6x^2 = 9 \implies x^2 = \frac{9}{6} = \frac{3}{2}\]
Solving for \(x\), we get \(x = \pm\sqrt{\frac{3}{2}} = \pm\frac{\sqrt{6}}{2}\). Since these values do not make the denominator zero, the x-intercepts are \((\frac{\sqrt{6}}{2}, 0)\) and \((-\frac{\sqrt{6}}{2}, 0)\).
- Finding the y-intercept
The y-intercept occurs where \(x = 0\):
\[f(0) = \frac{6(0)^2 - 9}{0^3 - 13} = \frac{-9}{-13} = \frac{9}{13}\]
Thus, the y-intercept is \((0, \frac{9}{13})\).
- Finding the vertical asymptote
The vertical asymptote occurs where the denominator is zero and the numerator is non-zero:
\[x^3 - 13 = 0 \implies x^3 = 13 \implies x = \sqrt[3]{13}\]
Therefore, the equation of the vertical asymptote is \(x = \sqrt[3]{13}\).
- Finding the horizontal asymptote
The horizontal asymptote is found by examining the limit of \(f(x)\) as \(x\) approaches infinity:
\[\lim_{x \to \infty} \frac{6x^2 - 9}{x^3 - 13} = 0\]
Since the degree of the denominator (3) is greater than the degree of the numerator (2), the function approaches 0 as \(x\) grows large. Therefore, the horizontal asymptote is \(y = 0\).
- Final Answer
In summary, for the function \(f(x) = \frac{6x^2 - 9}{x^3 - 13}\):
- \(f(9) = \frac{477}{716}\)
- x-intercepts: \(\left(\frac{\sqrt{6}}{2}, 0\right)\) and \(\left(-\frac{\sqrt{6}}{2}, 0\right)\)
- y-intercept: \(\left(0, \frac{9}{13}\right)\)
- Vertical asymptote: \(x = \sqrt[3]{13}\)
- Horizontal asymptote: \(y = 0\)
Examples
Understanding how to analyze rational functions is vital in fields like engineering and economics. For example, when designing an electrical filter, the vertical asymptotes of a transfer function (which is typically a ratio of polynomials) can indicate the resonant frequencies where the output might theoretically become infinite, signaling potential failure points or areas requiring dampening. Identifying these intercepts and asymptotes helps engineers design stable and efficient systems.