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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:
\[ \text{Domain: } (-\infty, \infty) \]

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### 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:
\[ \text{Domain: } [0, \infty) \]

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### 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:
\[ \text{Domain: } (-\infty, -2) \cup (-2, \infty) \]

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### 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:
\[ \text{Domain: } (-\infty, \infty) \]

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### 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:
\[ \text{Domain: } (-\infty, 0) \cup (0, \infty) \]

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### 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:
\[ \text{Domain: } [0, 3) \cup (3, \infty) \]

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### 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?

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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.

Calcular el dominio de las siguientes funciones.

Explore the domains of six different functions: three polynomial, two rational, and one radical. Learn how these domains are determined through algebraic analysis, including handling polynomial infinity domains and restrictions imposed by radicals and rational function denominators.

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