Verified Answer :
- Evaluate \(f(8) = \frac{1021}{492} \approx 2.0752\).
- Find x-intercepts by solving \(16x^2 - 3 = 0\), yielding \(\boxed{\left(-\frac{\sqrt{3}}{4}, 0\right), \left(\frac{\sqrt{3}}{4}, 0\right)}\).
- Determine y-intercept by calculating \(f(0) = \boxed{(0, 0.15)}\).
- Identify vertical asymptote at \(\boxed{x = \sqrt[3]{20}}\) and horizontal asymptote at \(\boxed{y = 0}\).
Explanation
- Function Analysis and Domain
To analyze the function \(f(x) = \frac{16x^2 - 3}{x^3 - 20}\), we examine its key properties: evaluation at a specific point, intercepts, and asymptotes. The domain consists of all real numbers except where the denominator is zero. The denominator \(x^3 - 20 = 0\) when \(x = \sqrt[3]{20} \approx 2.714\). Thus, the function is defined for all \(x eq \sqrt[3]{20}\).
- Evaluating f(8)
To evaluate \(f(8)\), we substitute \(x = 8\) into the function expression:
\[f(8) = \frac{16(8)^2 - 3}{8^3 - 20} = \frac{16(64) - 3}{512 - 20} = \frac{1024 - 3}{492} = \frac{1021}{492} \approx 2.0752\]
- Finding x-intercepts
The x-intercepts occur where the function value is zero, which happens when the numerator equals zero:
\[16x^2 - 3 = 0 \implies x^2 = \frac{3}{16} \implies x = \pm \frac{\sqrt{3}}{4}\]
Since \(\frac{\sqrt{3}}{4} \approx 0.433\), the x-intercepts are \(\left(-\frac{\sqrt{3}}{4}, 0\right)\) and \(\left(\frac{\sqrt{3}}{4}, 0\right)\).
- Finding the y-intercept
The y-intercept occurs where \(x = 0\):
\[f(0) = \frac{16(0)^2 - 3}{0^3 - 20} = \frac{-3}{-20} = 0.15\]
Thus, the y-intercept is \((0, 0.15)\).
- Identifying Asymptotes
Vertical asymptotes occur where the function is undefined and the numerator is non-zero, i.e., where \(x^3 - 20 = 0\). This gives the vertical asymptote \(x = \sqrt[3]{20}\).
For the horizontal asymptote, we examine the limit as \(x \to \pm \infty\). Since the degree of the denominator (3) is greater than the degree of the numerator (2), the function approaches 0 as \(x\) increases or decreases without bound. Thus, the horizontal asymptote is \(y = 0\).
Examples
In engineering, analyzing a function like this is essential for understanding how a system behaves as variables change. For instance, in modeling the temperature distribution along a rod or the pressure changes in a pipe system, identifying intercepts and asymptotes tells you where the value is zero, where it starts, and where it trends over time or distance, preventing potential system failures or identifying optimal operational ranges. The techniques used here, such as finding intercepts and limits for asymptotes, provide the foundation for robust design and analysis in various technical fields.