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To determine the domain and range of the logarithmic function [tex]\( f(x) \)[/tex] based on the given table, let’s walk through the analysis step-by-step.
### Analyzing the Function:
The table suggests that [tex]\( f(x) \)[/tex] is a logarithmic function, as seen from the tabulated values of [tex]\( x \)[/tex] and corresponding values of [tex]\( y \)[/tex]. The natural form of a logarithmic function is [tex]\( f(x) = \log_b(x) \)[/tex] for some base [tex]\( b \)[/tex].
Given:
[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:
- [tex]\( f(1) = 0 \)[/tex] implies the function intersects the x-axis at [tex]\( x = 1 \)[/tex].
- [tex]\( f\left(\frac{1}{125}\right) = -3 \)[/tex] implies when [tex]\( x = \frac{1}{125} \)[/tex], [tex]\( y = -3 \)[/tex].
- [tex]\( f(125) = 3 \)[/tex] implies when [tex]\( x = 125 \)[/tex], [tex]\( y = 3 \)[/tex].
These observations are consistent with the logarithmic function behavior.
### Domain:
The domain of a logarithmic function [tex]\( f(x) = \log_b(x) \)[/tex] for any base [tex]\( b \)[/tex] is all positive real numbers [tex]\( x > 0 \)[/tex]. This is because the logarithm of a non-positive number is undefined, and the logarithm of zero approaches negative infinity.
Expressed in notation:
- Inequality Notation: [tex]\( x > 0 \)[/tex]
- Interval Notation: [tex]\( (0, \infty) \)[/tex]
- Set-Builder Notation: [tex]\( \{x \in \mathbb{R} \mid x > 0\} \)[/tex]
### Range:
The range of a logarithmic function is all real numbers because [tex]\( \log_b(x) \)[/tex] can take any real value as [tex]\( x \)[/tex] varies over positive numbers.
Expressed in notation:
- Inequality Notation: There is no restriction needed since it covers all real numbers.
- Interval Notation: [tex]\( (-\infty, \infty) \)[/tex]
- Set-Builder Notation: [tex]\( \{y \in \mathbb{R}\} \)[/tex]
### Conclusion:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 0 \)[/tex].
- The range of [tex]\( f(x) \)[/tex] is all real numbers.
To summarize:
- Domain: [tex]\( x > 0 \)[/tex]
- Range: All real numbers
These findings align with our understanding of the properties of logarithmic functions. Now, we can use this information to graph the function [tex]\( f(x) = \log_b(x) \)[/tex], showing its behavior over the interval [tex]\( x > 0 \)[/tex], and extending infinitely in both the positive and negative [tex]\( y \)[/tex]-directions.
### Analyzing the Function:
The table suggests that [tex]\( f(x) \)[/tex] is a logarithmic function, as seen from the tabulated values of [tex]\( x \)[/tex] and corresponding values of [tex]\( y \)[/tex]. The natural form of a logarithmic function is [tex]\( f(x) = \log_b(x) \)[/tex] for some base [tex]\( b \)[/tex].
Given:
[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:
- [tex]\( f(1) = 0 \)[/tex] implies the function intersects the x-axis at [tex]\( x = 1 \)[/tex].
- [tex]\( f\left(\frac{1}{125}\right) = -3 \)[/tex] implies when [tex]\( x = \frac{1}{125} \)[/tex], [tex]\( y = -3 \)[/tex].
- [tex]\( f(125) = 3 \)[/tex] implies when [tex]\( x = 125 \)[/tex], [tex]\( y = 3 \)[/tex].
These observations are consistent with the logarithmic function behavior.
### Domain:
The domain of a logarithmic function [tex]\( f(x) = \log_b(x) \)[/tex] for any base [tex]\( b \)[/tex] is all positive real numbers [tex]\( x > 0 \)[/tex]. This is because the logarithm of a non-positive number is undefined, and the logarithm of zero approaches negative infinity.
Expressed in notation:
- Inequality Notation: [tex]\( x > 0 \)[/tex]
- Interval Notation: [tex]\( (0, \infty) \)[/tex]
- Set-Builder Notation: [tex]\( \{x \in \mathbb{R} \mid x > 0\} \)[/tex]
### Range:
The range of a logarithmic function is all real numbers because [tex]\( \log_b(x) \)[/tex] can take any real value as [tex]\( x \)[/tex] varies over positive numbers.
Expressed in notation:
- Inequality Notation: There is no restriction needed since it covers all real numbers.
- Interval Notation: [tex]\( (-\infty, \infty) \)[/tex]
- Set-Builder Notation: [tex]\( \{y \in \mathbb{R}\} \)[/tex]
### Conclusion:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 0 \)[/tex].
- The range of [tex]\( f(x) \)[/tex] is all real numbers.
To summarize:
- Domain: [tex]\( x > 0 \)[/tex]
- Range: All real numbers
These findings align with our understanding of the properties of logarithmic functions. Now, we can use this information to graph the function [tex]\( f(x) = \log_b(x) \)[/tex], showing its behavior over the interval [tex]\( x > 0 \)[/tex], and extending infinitely in both the positive and negative [tex]\( y \)[/tex]-directions.
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