To determine which type of parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] represents, let's analyze the function and the options provided.
1. Quadratic:
A quadratic function typically has the form [tex]\( f(x) = ax^2 + bx + c \)[/tex]. It involves a variable raised to the second power (squared). Since [tex]\( f(x) = \sqrt[3]{x} \)[/tex] does not involve squaring, [tex]\( f(x) \)[/tex] cannot be quadratic.
2. Cube root:
A cube root function is expressed as [tex]\( f(x) = \sqrt[3]{x} \)[/tex] or equivalently [tex]\( f(x) = x^{1/3} \)[/tex]. This function specifically involves taking the cube root of the variable [tex]\( x \)[/tex].
3. Reciprocal:
A reciprocal function has the form [tex]\( f(x) = \frac{1}{x} \)[/tex] or more generally [tex]\( f(x) = \frac{a}{x} \)[/tex], where the variable [tex]\( x \)[/tex] is in the denominator. The given function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] does not match this form.
4. Square root:
A square root function is written as [tex]\( f(x) = \sqrt{x} \)[/tex] or [tex]\( f(x) = x^{1/2} \)[/tex]. The variable [tex]\( x \)[/tex] is under a square root. Since our function [tex]\( f(x) \)[/tex] involves a cube root, it is different from the square root function.
Given the analysis:
- Quadratic: No
- Cube root: Yes
- Reciprocal: No
- Square root: No
Therefore, the correct type of parent function for [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is:
B. Cube root
