14.Find the domain of the function f ( x )

To find the domain of the function $f(x) = \frac{x^2 - 3x}{1}$, under the specified conditions, we need to determine the values of $x$ for which the function is defined.

The function is a rational expression, and it is valid everywhere except where the denominator is zero. However, since there is no denominator or problematic terms like square roots in this expression, the domain primarily depends on the specified intervals.

Analyzing the conditions:

1. $x > 0$: The function $f(x)$ is valid, and it remains defined.
2. $x < 0$ or $x > 3$: $f(x)$ remains defined here as well, as the numerator doesn’t impose restrictions.
3. $0 \leq x \leq 3$: The function is defined for $x$ values between 0 and 3, inclusive.
4. $x \leq 0$ or $x \geq 3$: This range specifies $x \leq 0$ (left side of zero) or $x \geq 3$ (right side of three), and the function remains defined here as well.
5. $x \geq 3$: The function is defined for $x \geq 3$.

The correct domain of the function depends on the range intervals provided. Based on the above information, the correct domain is:

$x \leq 0$ or $x \geq 3$.

Would you like additional details or explanations about the intervals?

Here are five related questions for deeper understanding:

1. How is the domain determined when a denominator is present in a function?
2. What happens to the domain of a function involving square roots?
3. How can the intersection of multiple domain constraints be calculated?
4. What are some examples of functions with discontinuities in their domains?
5. How does the range of a function relate to its domain?

Tip: Always check for constraints like division by zero or negative square roots when determining the domain of a function.