To determine which of the given functions is a parent function, let's analyze the essence of a parent function. A parent function is the simplest form of functions in a family, defining the primary characteristic of that function.
Let's review each option in detail:
A. [tex]\( f(x) = -3x^3 + 1 \)[/tex]
- A parent function for a cubic function typically looks like [tex]\( f(x) = x^3 \)[/tex].
- This function has a coefficient [tex]\(-3\)[/tex] and an additional constant term [tex]\( +1 \)[/tex], so it is not a parent function.
B. [tex]\( f(x) = 2^x \)[/tex]
- This function is an exponential function where [tex]\( x \)[/tex] is the exponent.
- The simplest form of an exponential function is of the form [tex]\( f(x) = a^x \)[/tex], where [tex]\( a \)[/tex] is a positive constant.
- Here, [tex]\( f(x) = 2^x \)[/tex] fits this form directly, so this is indeed a parent function.
C. [tex]\( f(x) = 3^x - \frac{x}{3} \)[/tex]
- This function is a combination of an exponential part [tex]\( 3^x \)[/tex] and a linear term [tex]\(-\frac{x}{3}\)[/tex].
- The presence of the linear term [tex]\(-\frac{x}{3}\)[/tex] disqualifies it from being a parent function because it introduces an additional term.
D. [tex]\( f(x) = \frac{1}{2} 3^x \)[/tex]
- This is an exponential function modified by a coefficient [tex]\(\frac{1}{2}\)[/tex].
- A parent function should not have any coefficients other than [tex]\(1\)[/tex]; therefore, this is not a parent function.
After analyzing each option, we find that the simplest form, which is the parent function, is:
The correct answer is:
B. [tex]\( f(x) = 2^x \)[/tex]
