Answer :
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.
