Answer :
To determine which of the following functions is a parent function, we first need to understand what a parent function is. A parent function is the simplest form of a given function family, typically without any transformations such as shifts, stretches, compressions, or flips.
Let's evaluate each option:
Option A: [tex]\( f(x) = 2 \cdot 3^x \)[/tex]
- This function involves an exponential base [tex]\( 3^x \)[/tex] but includes a multiplicative constant 2. The presence of this constant indicates it is not in its simplest form.
Option B: [tex]\( f(x) = 2 e^{2x} \)[/tex]
- This is an exponential function with the base [tex]\(e\)[/tex], but it includes both a multiplicative constant 2 and an exponent multiplication by 2. These transformations mean it is not in its simplest form.
Option C: [tex]\( f(x) = e^x \)[/tex]
- This function is an exponential function with base [tex]\(e\)[/tex]. There are no additional transformations or multiplicative constants. It is in its simplest form, making it a candidate for the parent function.
Option D: [tex]\( f(x) = x^4 + 3 \)[/tex]
- This function is a polynomial function of degree 4, but it includes an additional constant term 3. The simplest form of a polynomial function of degree 4 would be [tex]\(x^4\)[/tex]; thus, it is not in its simplest form.
After examining all the options, the correct answer is:
C. [tex]\( f(x) = e^x \)[/tex]
This is the parent function among the given choices, as it is in the simplest form for an exponential function with base [tex]\(e\)[/tex].
Let's evaluate each option:
Option A: [tex]\( f(x) = 2 \cdot 3^x \)[/tex]
- This function involves an exponential base [tex]\( 3^x \)[/tex] but includes a multiplicative constant 2. The presence of this constant indicates it is not in its simplest form.
Option B: [tex]\( f(x) = 2 e^{2x} \)[/tex]
- This is an exponential function with the base [tex]\(e\)[/tex], but it includes both a multiplicative constant 2 and an exponent multiplication by 2. These transformations mean it is not in its simplest form.
Option C: [tex]\( f(x) = e^x \)[/tex]
- This function is an exponential function with base [tex]\(e\)[/tex]. There are no additional transformations or multiplicative constants. It is in its simplest form, making it a candidate for the parent function.
Option D: [tex]\( f(x) = x^4 + 3 \)[/tex]
- This function is a polynomial function of degree 4, but it includes an additional constant term 3. The simplest form of a polynomial function of degree 4 would be [tex]\(x^4\)[/tex]; thus, it is not in its simplest form.
After examining all the options, the correct answer is:
C. [tex]\( f(x) = e^x \)[/tex]
This is the parent function among the given choices, as it is in the simplest form for an exponential function with base [tex]\(e\)[/tex].
