Question
How many solutions does a positive cubed root function have: 0, 1, or 2?
Answers
GPT-4o mini
A positive cubed root function has one solution.
The function \( f(x) = \sqrt[3]{x} \) is continuous and monotonically increasing for all real numbers \( x \). For any given positive value of \( y \), there exists exactly one \( x \) such that \( \sqrt[3]{x} = y \). Specifically, if you solve for \( x \) in the equation \( y = \sqrt[3]{x} \), you can rearrange it to \( x = y^3 \), which gives one solution for any positive \( y \).
Therefore, regardless of the values you provide within the domain of the cubic root function, you will only find one solution when you consider \( x \) values that yield positive outputs. Thus, the answer is **1 solution**.
The function \( f(x) = \sqrt[3]{x} \) is continuous and monotonically increasing for all real numbers \( x \). For any given positive value of \( y \), there exists exactly one \( x \) such that \( \sqrt[3]{x} = y \). Specifically, if you solve for \( x \) in the equation \( y = \sqrt[3]{x} \), you can rearrange it to \( x = y^3 \), which gives one solution for any positive \( y \).
Therefore, regardless of the values you provide within the domain of the cubic root function, you will only find one solution when you consider \( x \) values that yield positive outputs. Thus, the answer is **1 solution**.