Let's go through the problems one by one:

1. Domain of $f(x) = 3x^2 + 4$

The function $f(x) = 3x^2 + 4$ is a quadratic function, and quadratic functions have a domain of all real numbers. This is because there are no restrictions on the input $x$ (no division by zero or square roots of negative numbers).

Domain of $f(x)$: $(-\infty, \infty)$

2. Range of $f(x) = 3x^2 + 4$

This is a parabola opening upwards (since the coefficient of $x^2$ is positive). The minimum value of the function occurs when $x = 0$, giving us: $f(0) = 3(0)^2 + 4 = 4$ Since the parabola opens upwards, the range of $f(x)$ starts at 4 and goes to infinity.

Range of $f(x)$: $[4, \infty)$

3. Domain of $g(x) = \frac{\sqrt{x - 4}}{x - 10}$

For $g(x)$, we need to look at two restrictions:

• The square root function $\sqrt{x - 4}$ requires that $x - 4 \geq 0$, so $x \geq 4$.
• The denominator $x - 10$ cannot be zero, so $x \neq 10$.

Therefore, the domain is all $x \geq 4$ except $x = 10$.

Domain of $g(x)$: $[4, 10) \cup (10, \infty)$

4. Range of $g(x) = \frac{\sqrt{x - 4}}{x - 10}$

The function has a square root in the numerator and a linear expression in the denominator. As $x \to 4^+$, $g(x) \to 0$, and as $x \to \infty$, the behavior of the function will depend on the balance between the square root in the numerator and the linear term in the denominator. The function does not have a simple maximum or minimum, but we can deduce that the range is all real numbers except 0.

Range of $g(x)$: $(-\infty, 0) \cup (0, \infty)$

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Would you like more detailed explanations on any of these steps? Here are some related questions to explore:

1. What happens to the domain and range of $f(x)$ if you change the constant term (e.g., from +4 to -4)?
2. How would you graph the function $g(x) = \frac{\sqrt{x - 4}}{x - 10}$ to confirm its domain and range visually?
3. Can you find the horizontal asymptote of $g(x)$?
4. What would happen to the domain of $g(x)$ if we changed the denominator to $x + 10$?
5. How does the range of a function like $f(x) = 3x^2 + 4$ change if we add a negative sign in front of the $x^2$ term?

Tip: When dealing with square roots or denominators, always check the conditions for valid inputs (such as non-negative values for square roots and non-zero values for denominators).