To find the domain of the function $f(x) = \frac{x}{4} + 1$, we need to determine the set of all possible values of $x$ for which the function is defined.

Analysis of the Function

The function $f(x) = \frac{x}{4} + 1$ is a linear function. Linear functions are defined for all real numbers since there are no restrictions like division by zero or square roots of negative numbers.

Domain

The domain of $f(x) = \frac{x}{4} + 1$ is therefore all real numbers, which is typically represented as $(-\infty, \infty)$.

Provided Options:

Let's analyze the options given:

1. Option 1: $(-0,-4) (-4,0∞)$ - This is not a valid representation and seems to be a mistake.
2. Option 2: $(-0,4) \cup (4,\infty)$ - This represents two intervals excluding the values $x = 0$ and $x = 4$, which is incorrect because the function is defined for all $x$.
3. Option 3: $(-∞, 0) \cup (0,0)$ - This option is incorrect because the domain excludes $x = 0$ and does not cover all real numbers.
4. Option 4: $(18,8)$ - This option is not correct, as it suggests a single interval from 18 to 8, which is also written incorrectly (should be $(8,18)$).

None of these options correctly describe the domain of the function.

Correct Domain:

The correct domain is:

$\text{Domain: } (-\infty, \infty)$

Would you like more details, or do you have any other questions?

Related Questions:

1. What is the domain of $f(x) = \frac{1}{x - 2}$?
2. How do you find the domain of $f(x) = \sqrt{x + 5}$?
3. What is the effect of adding a constant to a function on its domain?
4. How do you determine the domain of a function with a square root and a fraction?
5. Can a linear function ever have a restricted domain?

Tip: The domain of a linear function $f(x) = ax + b$ is always all real numbers, $(-\infty, \infty)$, unless explicitly restricted by the problem context.