To evaluate the function $f(x) = \frac{8 - x}{3x^2 - 4}$ at the given values of $x$, we substitute $x = 2.5$ and $x = 2.01$ into the equation.

1. Evaluating $f(2.5)$:

$f(2.5) = \frac{8 - 2.5}{3(2.5)^2 - 4}$ Calculate the numerator and the denominator separately:

• Numerator: $8 - 2.5 = 5.5$
• Denominator: $3(2.5)^2 - 4 = 3(6.25) - 4 = 18.75 - 4 = 14.75$

Thus: $f(2.5) = \frac{5.5}{14.75} \approx 0.37288136$

2. Evaluating $f(2.01)$:

$f(2.01) = \frac{8 - 2.01}{3(2.01)^2 - 4}$ Calculate the numerator and the denominator separately:

• Numerator: $8 - 2.01 = 5.99$
• Denominator: $3(2.01)^2 - 4 = 3(4.0401) - 4 = 12.1203 - 4 = 8.1203$

Thus: $f(2.01) = \frac{5.99}{8.1203} \approx 0.73771277$

Answers:

(i) $f(2.5) \approx 0.37288136$

(ii) $f(2.01) \approx 0.73771277$

Would you like further details or have any questions?

Here are five additional related questions:

1. What is the domain of the function $f(x) = \frac{8 - x}{3x^2 - 4}$?
2. What happens to $f(x)$ as $x$ approaches 0?
3. Can we find the asymptotes of $f(x)$? If so, what are they?
4. How would you determine if the function has any points of discontinuity?
5. What is the behavior of $f(x)$ as $x \to \infty$?

Tip: When evaluating functions, always calculate the numerator and denominator separately for clarity and accuracy.