For each of the following functions, identify any restrictions on its domain.
$F(x)=\sqrt[3]{x-2}+3$
Is there any value of $x$ that would cause this function to be undefined?
If there are restrictions on the domain, explain those restrictions. If there are no restrictions, explain why that is.
Answer (2)
The function is F ( x ) = 3 x − 2 + 3 .
Cube roots are defined for all real numbers.
Therefore, there are no restrictions on the domain of F ( x ) .
The domain is all real numbers. No restrictions
Explanation
Analyzing the Function The given function is F ( x ) = 3 x − 2 + 3 . We need to determine if there are any restrictions on the domain of this function. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Considering the Cube Root The function involves a cube root, 3 x − 2 . Unlike square roots (or other even roots), cube roots are defined for all real numbers, including negative numbers and zero. This is because any real number has a unique cube root. For example, the cube root of -8 is -2, since ( − 2 ) 3 = − 8 .
Determining Restrictions Since the cube root is defined for all real numbers, the expression inside the cube root, x − 2 , can be any real number. There are no restrictions on the values that x can take. Therefore, there are no values of x that would make the function undefined.
Conclusion The domain of F ( x ) = 3 x − 2 + 3 is all real numbers because the cube root function is defined for all real numbers.
Examples
Understanding the domain of functions is crucial in many real-world applications. For example, when modeling the growth of a plant, the domain might represent time, which cannot be negative. Similarly, in physics, the domain of a function describing the trajectory of a projectile might be limited by the physical constraints of the environment. In this case, since the cube root function has no restrictions, it could model scenarios where negative values are meaningful, such as temperature variations below zero degrees Celsius.
The function F ( x ) = 3 x − 2 + 3 has no restrictions on its domain. This is because the cube root function is defined for all real numbers. Therefore, the domain is all real numbers, or ( − ∞ , ∞ ) .
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