Question
nsider the function \( f(x)=-(x-2)^{3}+8 \). What is the domain of this function? (1 point) \( (2, \infty) \) \( (-\infty, 2) \) \( (-\infty, \infty) \) \( (-\infty, 2) \)
Ask by Lowe Mitchell. in the United States
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The domain of the function \( f(x) = -(x-2)^{3} + 8 \) is all real numbers, so the correct answer is \( (-\infty, \infty) \).
Solution
To determine the domain of the function \( f(x) = -(x-2)^{3} + 8 \), we need to analyze the expression.
1. **Identify the type of function**: The function is a polynomial function because it is a cubic polynomial (the term \( (x-2)^3 \) is raised to the power of 3).
2. **Properties of polynomial functions**: Polynomial functions are defined for all real numbers. This means there are no restrictions on the values that \( x \) can take.
3. **Conclusion**: Since there are no restrictions, the domain of the function \( f(x) \) is all real numbers.
Thus, the domain of the function is \( (-\infty, \infty) \).
The correct answer is:
\( (-\infty, \infty) \)
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Simplify this solution Mind Expander
The function \( f(x)=-(x-2)^{3}+8 \) is a polynomial function, and polynomial functions have a domain of all real numbers. This means that you can plug in any real value for \( x \) without running into issues like division by zero or taking the square root of a negative number. So, the correct answer is \( (-\infty, \infty) \). Additionally, polynomial functions are known for their smooth curves and continuous nature, indicating there are no gaps or breaks in their graphs. For \( f(x) \), as \( x \) approaches positive or negative infinity, the function will also approach infinity, exhibiting the usual behavior of cubic functions influenced by translation and reflection.
