To determine which of the options given represents a parent function, we need to understand what a parent function is. A parent function is the simplest form of a function type, without any transformations such as shifts, stretches, compressions, or reflections.
Let's analyze each option:
A. [tex]\( f(x) = 3^x - \frac{x}{3} \)[/tex]
- This is a combination of an exponential function, [tex]\( 3^x \)[/tex], and a linear term, [tex]\( -\frac{x}{3} \)[/tex]. Since a parent function should not have additional linear terms or modifications, this is not a parent function.
B. [tex]\( f(x) = -3x^3 + 1 \)[/tex]
- This function is a cubic function [tex]\( x^3 \)[/tex] but it has been modified by multiplying by -3 and shifting up by 1 unit. Thus, it is not the simplest form of a cubic function and hence not a parent function.
C. [tex]\( f(x) = \frac{1}{2} \cdot 3^x \)[/tex]
- This is an exponential function that is being scaled by [tex]\(\frac{1}{2}\)[/tex]. Scaling the function modifies the basic form. Therefore, this is not a parent function.
D. [tex]\( f(x) = 2^x \)[/tex]
- This is a basic exponential function with the base 2. It has not been shifted, scaled, or translated in any way. It represents the simplest form of an exponential function. Therefore, this is a parent function.
Therefore, the correct answer is:
D. [tex]\( f(x) = 2^x \)[/tex]
