Let's analyze and calculate the domain for each of the given functions:

a) $f(x) = x^3 + 3x - 4$

This is a cubic polynomial, and polynomials are defined for all real numbers. Therefore, the domain is: $\text{Domain: } (-\infty, \infty)$

b) $g(t) = t^{1/4}$

This represents the fourth root of $t$. The function is only defined when the argument of the root is non-negative, i.e., $t \geq 0$.

So, the domain is: $\text{Domain: } [0, \infty)$

c) $f(t) = \frac{1}{t+2}$

The function is undefined when the denominator is zero. So, $t + 2 = 0$, which gives $t = -2$. The function is defined for all real numbers except $t = -2$.

Thus, the domain is: $\text{Domain: } (-\infty, -2) \cup (-2, \infty)$

d) $f(t) = \frac{1}{t^2 + 4}$

The denominator $t^2 + 4$ is always positive (since $t^2 \geq 0$ and adding 4 makes it strictly positive), so the function is defined for all real numbers.

Thus, the domain is: $\text{Domain: } (-\infty, \infty)$

e) $h(z) = z^3 + z^{-3} = z^3 + \frac{1}{z^3}$

The function will be undefined when $z^3 = 0$, which means $z = 0$. Therefore, the function is defined for all real numbers except $z = 0$.

Thus, the domain is: $\text{Domain: } (-\infty, 0) \cup (0, \infty)$

f) $f(x) = \frac{\sqrt{x}}{x^2 - 9}$

For this function, two conditions need to be considered:

1. The square root $\sqrt{x}$ requires that $x \geq 0$.
2. The denominator $x^2 - 9$ must not be zero, so $x^2 \neq 9$, which gives $x \neq \pm 3$.

Thus, the domain is: $\text{Domain: } [0, 3) \cup (3, \infty)$

Summary of domains:

• a) $(-\infty, \infty)$
• b) $[0, \infty)$
• c) $(-\infty, -2) \cup (-2, \infty)$
• d) $(-\infty, \infty)$
• e) $(-\infty, 0) \cup (0, \infty)$
• f) $[0, 3) \cup (3, \infty)$

Do you have any questions or want details about a specific part?

Here are some related questions for further practice:

1. What is the domain of $f(x) = \frac{1}{x^2 - 1}$?
2. How do you find the domain of a logarithmic function?
3. What is the domain of $g(x) = \sqrt{x - 2}$?
4. How do you determine the domain of a rational function?
5. What is the domain of $h(x) = \frac{x + 1}{x - 4}$?

Tip: Always check both the square roots and denominators when determining the domain of a function.