IDNLearn.com makes it easy to find accurate answers to your specific questions. Our experts provide timely and precise responses to help you understand and solve any issue you face.
Sagot :
Let's analyze the given table and determine the domain and range of the logarithmic function [tex]\( f(x) \)[/tex].
### Understanding the Table
The table shows values of [tex]\( x \)[/tex] and their corresponding [tex]\( y \)[/tex] values for the function [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline \frac{1}{125} & -3 \\ \hline \frac{1}{25} & -2 \\ \hline \frac{1}{5} & -1 \\ \hline 1 & 0 \\ \hline 5 & 1 \\ \hline 25 & 2 \\ \hline 125 & 3 \\ \hline \end{array} \][/tex]
From the table, we can observe that [tex]\( y = f(x) \)[/tex] corresponds to the logarithm of [tex]\( x \)[/tex] values. Specifically, this aligns with the function [tex]\( f(x) = \log_b(x) \)[/tex] where [tex]\( b \)[/tex] is a constant base.
### Determining the Domain of [tex]\( f(x) \)[/tex]
The domain of a logarithmic function [tex]\( \log_b(x) \)[/tex] is determined by where the argument [tex]\( x \)[/tex] is positive since the logarithm is only defined for positive numbers.
By examining the [tex]\( x \)[/tex]-values in the table:
[tex]\[ \left\{ \frac{1}{125}, \frac{1}{25}, \frac{1}{5}, 1, 5, 25, 125 \right\} \][/tex]
We see that all the [tex]\( x \)[/tex]-values are positive. Hence, the domain of [tex]\( f(x) \)[/tex] is all positive real numbers.
Domain in inequality notation:
[tex]\[ x > 0 \][/tex]
Domain in interval notation:
[tex]\[ (0, \infty) \][/tex]
### Determining the Range of [tex]\( f(x) \)[/tex]
The range of a logarithmic function typically spans all real numbers, as [tex]\( y \)[/tex]-values can take any real number based on the argument [tex]\( x \)[/tex]. However, since we are given a specific set of values for [tex]\( y \)[/tex]:
[tex]\[ \left\{ -3, -2, -1, 0, 1, 2, 3 \right\} \][/tex]
It indicates that [tex]\( f(x) \)[/tex] specifically maps to these values within this context.
Range in set-builder notation:
[tex]\[ \{ y \mid y \text{ is an integer and } -3 \le y \le 3 \} \][/tex]
### Summary
Based on the given table and the analysis:
- Domain of [tex]\( f(x) \)[/tex]:
[tex]\[ x > 0 \quad \text{or} \quad (0, \infty) \][/tex]
- Range of [tex]\( f(x) \)[/tex]:
[tex]\[ \{ y \mid y \text{ is an integer and } -3 \le y \le 3 \} \][/tex]
By understanding the table and the properties of the logarithmic function, we determined the domain and range precisely, aligning with the typical behavior of logarithmic functions.
### Understanding the Table
The table shows values of [tex]\( x \)[/tex] and their corresponding [tex]\( y \)[/tex] values for the function [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline \frac{1}{125} & -3 \\ \hline \frac{1}{25} & -2 \\ \hline \frac{1}{5} & -1 \\ \hline 1 & 0 \\ \hline 5 & 1 \\ \hline 25 & 2 \\ \hline 125 & 3 \\ \hline \end{array} \][/tex]
From the table, we can observe that [tex]\( y = f(x) \)[/tex] corresponds to the logarithm of [tex]\( x \)[/tex] values. Specifically, this aligns with the function [tex]\( f(x) = \log_b(x) \)[/tex] where [tex]\( b \)[/tex] is a constant base.
### Determining the Domain of [tex]\( f(x) \)[/tex]
The domain of a logarithmic function [tex]\( \log_b(x) \)[/tex] is determined by where the argument [tex]\( x \)[/tex] is positive since the logarithm is only defined for positive numbers.
By examining the [tex]\( x \)[/tex]-values in the table:
[tex]\[ \left\{ \frac{1}{125}, \frac{1}{25}, \frac{1}{5}, 1, 5, 25, 125 \right\} \][/tex]
We see that all the [tex]\( x \)[/tex]-values are positive. Hence, the domain of [tex]\( f(x) \)[/tex] is all positive real numbers.
Domain in inequality notation:
[tex]\[ x > 0 \][/tex]
Domain in interval notation:
[tex]\[ (0, \infty) \][/tex]
### Determining the Range of [tex]\( f(x) \)[/tex]
The range of a logarithmic function typically spans all real numbers, as [tex]\( y \)[/tex]-values can take any real number based on the argument [tex]\( x \)[/tex]. However, since we are given a specific set of values for [tex]\( y \)[/tex]:
[tex]\[ \left\{ -3, -2, -1, 0, 1, 2, 3 \right\} \][/tex]
It indicates that [tex]\( f(x) \)[/tex] specifically maps to these values within this context.
Range in set-builder notation:
[tex]\[ \{ y \mid y \text{ is an integer and } -3 \le y \le 3 \} \][/tex]
### Summary
Based on the given table and the analysis:
- Domain of [tex]\( f(x) \)[/tex]:
[tex]\[ x > 0 \quad \text{or} \quad (0, \infty) \][/tex]
- Range of [tex]\( f(x) \)[/tex]:
[tex]\[ \{ y \mid y \text{ is an integer and } -3 \le y \le 3 \} \][/tex]
By understanding the table and the properties of the logarithmic function, we determined the domain and range precisely, aligning with the typical behavior of logarithmic functions.
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.
