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Let's solve each part of the problem, step by step.

### (a) \( f(x) = \sqrt{64 - x} \)

For a square root function, the expression inside the square root must be non-negative, as the square root of a negative number is not defined in the set of real numbers. So, we need to solve:

\[
64 - x \geq 0
\]

Solving this inequality:

\[
x \leq 64
\]

Thus, the domain of \( f(x) \) is \( (-\infty, 64] \).

### (b) \( f(x) = \frac{x + 9}{8x^2 - x - 9} \)

For a rational function, the denominator cannot be zero, so we need to find when the denominator equals zero and exclude those values from the domain. Set the denominator equal to zero:

\[
8x^2 - x - 9 = 0
\]

We need to solve this quadratic equation. Using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(8)(-9)}}{2(8)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 288}}{16} = \frac{1 \pm \sqrt{289}}{16} = \frac{1 \pm 17}{16}
\]

This gives two solutions:

\[
x = \frac{1 + 17}{16} = \frac{18}{16} = \frac{9}{8}, \quad x = \frac{1 - 17}{16} = \frac{-16}{16} = -1
\]

So, the denominator is zero at \( x = \frac{9}{8} \) and \( x = -1 \). We must exclude these values from the domain.

Thus, the domain is:

\[
(-\infty, -1) \cup (-1, \frac{9}{8}) \cup (\frac{9}{8}, \infty)
\]

### Summary of Domains:
- (a) Domain: \( (-\infty, 64] \)
- (b) Domain: \( (-\infty, -1) \cup (-1, \frac{9}{8}) \cup (\frac{9}{8}, \infty) \)

Would you like further details or explanations on any part?

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Here are 5 related questions to further explore:
1. How do you solve inequalities involving square roots?
2. Can we have domains that are restricted to integers?
3. How do you factor quadratic expressions efficiently?
4. What is the significance of domain and range in functions?
5. How can we visualize the domain and range on a graph?

**Tip**: Always check for restrictions in both the numerator (if any square roots or logarithms) and the denominator of a function to determine the domain.

Find the domain of the function. (a) f(x) = √(64 - x) (b) f(x) = (x + 9) / (8x^2 - x - 9).

Learn how to find the domain of a square root function f(x) = √(64 - x) and a rational function f(x) = (x + 9) / (8x^2 - x - 9) with step-by-step solutions. This problem covers key algebraic concepts such as domain restrictions for square roots and rational expressions, as well as solving quadratic equations. Suitable for students in Grades 9-11.

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