Answer :
To determine which of the following functions is a parent function, we need to understand the concept of a parent function. A parent function is the simplest function of a family of functions that preserves the definition and basic shape of the function.
Let's analyze each option:
A. \( f(x) = e^x \)
This function is an exponential function where the base is \( e \). It is in its simplest form without any transformations such as scaling, translating, or reflecting. Hence, \( f(x) = e^x \) is indeed a parent function.
B. \( f(x) = 2e^{2x} \)
This function is also an exponential function, but it includes transformations. Specifically, it has a vertical stretch by a factor of 2 and a horizontal compression by a factor of \( \frac{1}{2} \). These transformations indicate that this function is derived from the parent function \( e^x \).
C. \( f(x) = x^4 + 3 \)
This is a polynomial function, but it includes a translation upward by 3 units. The parent function in this family would be \( f(x) = x^4 \), which is the simplest form without any translations or other transformations. Therefore, \( x^4 + 3 \) is not a parent function.
D. \( f(x) = 2 \cdot 3^x \)
This function is an exponential function with the base 3, but with a vertical stretch by a factor of 2. The parent function here would be \( 3^x \), without any additional multipliers or transformations. Hence, \( 2 \cdot 3^x \) is not a parent function.
Based on this analysis, the parent function among the given options is:
[tex]\[ A. \, f(x) = e^x \][/tex]
Let's analyze each option:
A. \( f(x) = e^x \)
This function is an exponential function where the base is \( e \). It is in its simplest form without any transformations such as scaling, translating, or reflecting. Hence, \( f(x) = e^x \) is indeed a parent function.
B. \( f(x) = 2e^{2x} \)
This function is also an exponential function, but it includes transformations. Specifically, it has a vertical stretch by a factor of 2 and a horizontal compression by a factor of \( \frac{1}{2} \). These transformations indicate that this function is derived from the parent function \( e^x \).
C. \( f(x) = x^4 + 3 \)
This is a polynomial function, but it includes a translation upward by 3 units. The parent function in this family would be \( f(x) = x^4 \), which is the simplest form without any translations or other transformations. Therefore, \( x^4 + 3 \) is not a parent function.
D. \( f(x) = 2 \cdot 3^x \)
This function is an exponential function with the base 3, but with a vertical stretch by a factor of 2. The parent function here would be \( 3^x \), without any additional multipliers or transformations. Hence, \( 2 \cdot 3^x \) is not a parent function.
Based on this analysis, the parent function among the given options is:
[tex]\[ A. \, f(x) = e^x \][/tex]
