Math Problem Statement

Calcular el dominio de las siguientes funciones.

Solution

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

a) f(x)=x3+3x4f(x) = x^3 + 3x - 4

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


b) g(t)=t1/4g(t) = t^{1/4}

This represents the fourth root of tt. The function is only defined when the argument of the root is non-negative, i.e., t0t \geq 0.

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


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

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

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


d) f(t)=1t2+4f(t) = \frac{1}{t^2 + 4}

The denominator t2+4t^2 + 4 is always positive (since t20t^2 \geq 0 and adding 4 makes it strictly positive), so the function is defined for all real numbers.

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


e) h(z)=z3+z3=z3+1z3h(z) = z^3 + z^{-3} = z^3 + \frac{1}{z^3}

The function will be undefined when z3=0z^3 = 0, which means z=0z = 0. Therefore, the function is defined for all real numbers except z=0z = 0.

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


f) f(x)=xx29f(x) = \frac{\sqrt{x}}{x^2 - 9}

For this function, two conditions need to be considered:

  1. The square root x\sqrt{x} requires that x0x \geq 0.
  2. The denominator x29x^2 - 9 must not be zero, so x29x^2 \neq 9, which gives x±3x \neq \pm 3.

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


Summary of domains:

  • a) (,)(-\infty, \infty)
  • b) [0,)[0, \infty)
  • c) (,2)(2,)(-\infty, -2) \cup (-2, \infty)
  • d) (,)(-\infty, \infty)
  • e) (,0)(0,)(-\infty, 0) \cup (0, \infty)
  • f) [0,3)(3,)[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)=1x21f(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)=x2g(x) = \sqrt{x - 2}?
  4. How do you determine the domain of a rational function?
  5. What is the domain of h(x)=x+1x4h(x) = \frac{x + 1}{x - 4}?

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

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Rational Functions
Radical Functions

Formulas

f(x) = x^3 + 3x - 4
g(t) = t^{1/4}
f(t) = 1/(t+2)
f(t) = 1/(t^2 + 4)
h(z) = z^3 + z^{-3}
f(x) = sqrt(x) / (x^2 - 9)

Theorems

Domain Determination of Polynomials
Domain Determination of Radical Functions
Domain Determination of Rational Functions

Suitable Grade Level

High School