Math Problem Statement
Solution
Let's analyze and calculate the domain for each of the given functions:
a)
This is a cubic polynomial, and polynomials are defined for all real numbers. Therefore, the domain is:
b)
This represents the fourth root of . The function is only defined when the argument of the root is non-negative, i.e., .
So, the domain is:
c)
The function is undefined when the denominator is zero. So, , which gives . The function is defined for all real numbers except .
Thus, the domain is:
d)
The denominator is always positive (since and adding 4 makes it strictly positive), so the function is defined for all real numbers.
Thus, the domain is:
e)
The function will be undefined when , which means . Therefore, the function is defined for all real numbers except .
Thus, the domain is:
f)
For this function, two conditions need to be considered:
- The square root requires that .
- The denominator must not be zero, so , which gives .
Thus, the domain is:
Summary of domains:
- a)
- b)
- c)
- d)
- e)
- f)
Do you have any questions or want details about a specific part?
Here are some related questions for further practice:
- What is the domain of ?
- How do you find the domain of a logarithmic function?
- What is the domain of ?
- How do you determine the domain of a rational function?
- What is the domain of ?
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
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