To determine which of the following functions is a parent function, we need to understand what a parent function is. A parent function is the simplest function in a family of functions that preserves the definition or shape of the entire family. Parent functions serve as the template from which a family of functions is derived through transformations like translations, stretches, compressions, and reflections.
Let's analyze each of the given options:
A. [tex]\( f(x) = e^{3x} - 2 \)[/tex]
- This is an exponential function with a base of [tex]\( e \)[/tex], which has been horizontally scaled by a factor of 3 and then shifted downward by 2 units. While [tex]\( e^x \)[/tex] is a parent function in the family of exponential functions, [tex]\( e^{3x} - 2 \)[/tex] itself is not a parent function because of these transformations.
B. [tex]\( f(x) = -2^{x+1} \)[/tex]
- This is an exponential function with a base of 2, which has been translated horizontally and reflected across the x-axis. Like in option A, [tex]\( 2^x \)[/tex] is the parent function in the family of exponential functions, but [tex]\( -2^{x+1} \)[/tex] is not due to its transformations.
C. [tex]\( f(x) = 4 \)[/tex]
- This is a constant function since it returns the value 4 for all [tex]\( x \)[/tex]. The parent function of the constant function family is simply [tex]\( f(x) = c \)[/tex] where [tex]\( c \)[/tex] is any constant. Therefore, [tex]\( f(x) = 4 \)[/tex] is considered a parent function in the family of constant functions.
D. [tex]\( f(x) = 4x^2 \)[/tex]
- This is a quadratic function which has been vertically scaled by a factor of 4. The parent function of the quadratic function family is [tex]\( f(x) = x^2 \)[/tex]. Since [tex]\( 4x^2 \)[/tex] involves a vertical stretch, it is not the simplest form and thus not a parent function.
Based on this analysis, the function that is considered a parent function is:
C. [tex]\( f(x) = 4 \)[/tex]
