Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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]
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.