Given F(x) = x^(1/3) and g(x) = x^(2/3), determine the domain and range of each function.
Understand the Problem
The question appears to focus on the functions F(x) and g(x), specifically addressing their domain and range, as well as evaluating F(x) at a specific point. It also includes the representation of these functions and some calculations related to them.
Answer
Dom(F(x)) = $\mathbb{R}$; Dom(g(x)) = $\mathbb{R}$; Ran(g(x)) = $[0, \infty)$; $F(3) = \sqrt[3]{3}$.
Answer for screen readers
- Dom(F(x)) = $\mathbb{R}$
- Dom(g(x)) = $\mathbb{R}$
- Ran(g(x)) = $[0, \infty)$
- $F(3) = \sqrt[3]{3}$
Steps to Solve
-
Identify the Domain of F(x)
The function $F(x) = x^{1/3}$ is defined for all real numbers. Therefore, the domain is: $$ \text{Dom}(F(x)) = \mathbb{R} $$ -
Identify the Domain of g(x)
The function appears to be defined in a way that is not fully visible, but if we assume $g(x) = x^{2/3}$, then $g(x)$ is also defined for all real numbers: $$ \text{Dom}(g(x)) = \mathbb{R} $$ -
Identify the Range of g(x)
To find the range of ( g(x) ), we consider ( g(x) = x^{2/3} ). Since the output is non-negative for all real inputs, the range is: $$ \text{Ran}(g(x)) = [0, \infty) $$ -
Evaluate F(x) at a Specific Point
The problem mentions calculating $F(x)$, specifically at $F(3)$. We substitute ( x = 3 ): $$ F(3) = 3^{1/3} = \sqrt[3]{3} $$ -
Summarize Results
The overall results are:
- Domain of ( F(x) ): ( \mathbb{R} )
- Domain of ( g(x) ): ( \mathbb{R} )
- Range of ( g(x) ): ( [0, \infty) )
- Value of ( F(3) ): ( \sqrt[3]{3} )
- Dom(F(x)) = $\mathbb{R}$
- Dom(g(x)) = $\mathbb{R}$
- Ran(g(x)) = $[0, \infty)$
- $F(3) = \sqrt[3]{3}$
More Information
The cube root function $F(x) = x^{1/3}$ allows all real numbers as inputs and outputs since every real number has a cube root. The function $g(x) = x^{2/3}$ also allows all real numbers but only outputs non-negative values. Evaluating $F(3)$ gives us the cube root of 3, which can be approximated numerically as about 1.442.
Tips
- Failing to recognize that the cube and cube root functions can take any real number as input.
- Confusing the range of $g(x)$ with $F(x)$; remember that $g(x)$ outputs non-negative values only.
- Making computational errors when evaluating $F(3)$, especially if done without a calculator.
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