IDNLearn.com provides a seamless experience for finding accurate answers. Get thorough and trustworthy answers to your queries from our extensive network of knowledgeable professionals.
Sagot :
To determine the type of function that best describes [tex]\( g(x) \)[/tex] given the table of values, we proceed as follows:
[tex]\[ \begin{array}{|c|c|c|} \hline x & g(x) \\ \hline -3 & 4 \\ -1 & -4 \\ 1 & -4 \\ 3 & 4 \\ 5 & 20 \\ 7 & 44 \\ 9 & 76 \\ \hline \end{array} \][/tex]
We are asked to determine whether the function [tex]\( g(x) \)[/tex] is Exponential, Logarithmic, Polynomial, or Rational. Let's analyze each possibility:
1. Exponential Function:
- An exponential function generally has the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
- Exponential functions typically exhibit rapid growth (or decay). The values of [tex]\( g(x) \)[/tex] change sign and do not exhibit the consistent multiplicative growth characteristic of exponential functions.
2. Logarithmic Function:
- A logarithmic function has the form [tex]\( f(x) = a \cdot \log_b(x) + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
- Logarithmic functions usually grow more slowly and have a characteristic curve which is not reflected in the given values of [tex]\( g(x) \)[/tex]. Moreover, logarithmic functions are undefined for negative [tex]\( x \)[/tex]-values without considering transformations, which is not suitable given the mixed signs and values in the dataset.
3. Polynomial Function:
- A polynomial function has the form [tex]\( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex].
- Polynomial functions can describe various patterns of data changes, including changing directions multiple times (as seen with varying signs and values of [tex]\( g(x) \)[/tex]). The given values appearing for [tex]\( g(x) \)[/tex] exhibit behavior that might fit a polynomial of a particular degree.
4. Rational Function:
- A rational function is the ratio of two polynomials, [tex]\( f(x) = \frac{p(x)}{q(x)} \)[/tex].
- While rational functions can describe a variety of trends, including discontinuities and complex behaviors, the specific values given in [tex]\( g(x) \)[/tex] don’t immediately suggest a typical rational function’s behavior given more conventional fits for polynomial functions.
Through these analyses and matching characteristics:
From the given table and after examining the values, it can be inferred that the function that best fits the provided data pattern is a polynomial function.
Thus, the type of function that describes [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{\text{Polynomial}} \][/tex]
[tex]\[ \begin{array}{|c|c|c|} \hline x & g(x) \\ \hline -3 & 4 \\ -1 & -4 \\ 1 & -4 \\ 3 & 4 \\ 5 & 20 \\ 7 & 44 \\ 9 & 76 \\ \hline \end{array} \][/tex]
We are asked to determine whether the function [tex]\( g(x) \)[/tex] is Exponential, Logarithmic, Polynomial, or Rational. Let's analyze each possibility:
1. Exponential Function:
- An exponential function generally has the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
- Exponential functions typically exhibit rapid growth (or decay). The values of [tex]\( g(x) \)[/tex] change sign and do not exhibit the consistent multiplicative growth characteristic of exponential functions.
2. Logarithmic Function:
- A logarithmic function has the form [tex]\( f(x) = a \cdot \log_b(x) + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
- Logarithmic functions usually grow more slowly and have a characteristic curve which is not reflected in the given values of [tex]\( g(x) \)[/tex]. Moreover, logarithmic functions are undefined for negative [tex]\( x \)[/tex]-values without considering transformations, which is not suitable given the mixed signs and values in the dataset.
3. Polynomial Function:
- A polynomial function has the form [tex]\( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex].
- Polynomial functions can describe various patterns of data changes, including changing directions multiple times (as seen with varying signs and values of [tex]\( g(x) \)[/tex]). The given values appearing for [tex]\( g(x) \)[/tex] exhibit behavior that might fit a polynomial of a particular degree.
4. Rational Function:
- A rational function is the ratio of two polynomials, [tex]\( f(x) = \frac{p(x)}{q(x)} \)[/tex].
- While rational functions can describe a variety of trends, including discontinuities and complex behaviors, the specific values given in [tex]\( g(x) \)[/tex] don’t immediately suggest a typical rational function’s behavior given more conventional fits for polynomial functions.
Through these analyses and matching characteristics:
From the given table and after examining the values, it can be inferred that the function that best fits the provided data pattern is a polynomial function.
Thus, the type of function that describes [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{\text{Polynomial}} \][/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.
