Suppose $f(x)=\frac{16 x^2-3}{x^3-20}$.
The function is given by \(f(x) = \frac{16x^2 - 3}{x^3 - 20}\). To evaluate \(f(8)\), we substitute \(x = 8\) into the function:
\[f(8) = \frac{16(8)^2 - 3}{(8)^3 - 20}\]
First, calculate the numerator:
\[16(8^2) - 3 = 16(64) - 3 = 1024 - 3 = 1021\]
Next, calculate the denominator:
\[8^3 - 20 = 512 - 20 = 492\]
Thus,
\[f(8) = \frac{1021}{492} \approx 2.0752\]
The calculated value matches the provided information.
To find the x-intercepts of \(f(x)\), we set the function equal to zero, which occurs when the numerator is equal to zero and the denominator is non-zero.
Set the numerator to zero:
\[16x^2 - 3 = 0\]
\[16x^2 = 3\]
\[x^2 = \frac{3}{16}\]
Taking the square root of both sides gives:
\[x = \pm \sqrt{\frac{3}{16}} = \pm \frac{\sqrt{3}}{4}\]
We must verify these values do not make the denominator zero. The denominator is zero when \(x^3 - 20 = 0\), or \(x = \sqrt[3]{20}\). Since \(\frac{\sqrt{3}}{4} \approx 0.433\) and \(-\frac{\sqrt{3}}{4} \approx -0.433\), neither is equal to \(\sqrt[3]{20} \approx 2.714\).
Therefore, the x-intercepts as ordered pairs are \(\left(\frac{\sqrt{3}}{4}, 0\right)\) and \(\left(-\frac{\sqrt{3}}{4}, 0\right)\).
The x-intercepts represent the points where the graph of a function crosses the x-axis, which is essential in finding the roots of a physical system. For example, if a function describes the vertical displacement of an object over time, setting the function to zero allows you to determine exactly when the object hits the ground. Understanding these mathematical points helps engineers and scientists model motion, growth, and other dynamic phenomena accurately.