To find the domain of the function $f(x) = \frac{1}{3x + 8}$, we need to identify the values of $x$ for which the function is undefined. The function will be undefined when the denominator equals zero because division by zero is not possible.

Step 1: Find the value where the denominator is zero

Set the denominator equal to zero and solve for $x$: $3x + 8 = 0$

Step 2: Solve for $x$

$3x = -8$ $x = -\frac{8}{3}$

Step 3: Determine the domain

The function is defined for all real values of $x$ except $x = -\frac{8}{3}$.

Final Answer:

The only value not in the domain is $x = -\frac{8}{3}$.

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Here are 5 related questions that expand on this topic:

1. How do you find the domain of functions with square roots?
2. What is the domain of the function $g(x) = \sqrt{3x+8}$?
3. How does the domain change for functions like $f(x) = \frac{1}{(3x+8)^2}$?
4. What happens to the domain if the function is $f(x) = \ln(3x+8)$?
5. How do you determine the range of the function $f(x) = \frac{1}{3x+8}$?

Tip: When finding the domain of rational functions, always start by identifying values that make the denominator zero, as these are the points where the function is undefined.