Answer :
### Graphing the Function
To graph the logarithmic function [tex]\( f(x) \)[/tex] with the given points, you can plot the points [tex]\((x,y)\)[/tex] on a suitable coordinate plane. The given points are:
[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]
When you plot these points, you will observe that they lie on a curve typical of a logarithmic function. The characteristic shape of a logarithmic function is such that it increases slowly for small values of [tex]\( x \)[/tex], passes through the point (1, 0), and increases more rapidly as [tex]\( x \)[/tex] grows larger.
### Steps to Plot the Points
1. Plot the points [tex]\((\frac{1}{125}, -3)\)[/tex], [tex]\((\frac{1}{25}, -2)\)[/tex], [tex]\((\frac{1}{5}, -1)\)[/tex], [tex]\((1, 0)\)[/tex], [tex]\((5, 1)\)[/tex], [tex]\((25, 2)\)[/tex], and [tex]\((125, 3)\)[/tex].
2. Use a smooth curve to connect these points, ensuring that the curve passes through each plotted point.
3. Label your axes. Typically, it is useful to label the x-axis with a logarithmic scale in this case for better visualization since x-values span a broad range from [tex]\(\frac{1}{125}\)[/tex] to [tex]\(125\)[/tex].
### Determining the Domain
The logarithmic function [tex]\( f(x) = \log_b(x) \)[/tex] (for any base [tex]\( b \)[/tex]) is only defined for [tex]\( x > 0 \)[/tex]. This is because you cannot take the logarithm of a non-positive number.
Thus, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ x \in (0, \infty) \][/tex]
### Determining the Range
For the given [tex]\( y \)[/tex]-values, we observe that they span from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex]. However, logarithmic functions are not restricted to this specific interval; they can take any real number value given appropriate [tex]\( x \)[/tex]-values.
Therefore, for the general logarithmic function [tex]\( f(x) \)[/tex], the range is:
[tex]\[ y \in (-\infty, \infty) \][/tex]
### Conclusion
In summary:
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
By recognizing the nature of logarithmic functions and examining the given data points, you can accurately graph [tex]\( f(x) \)[/tex] and determine its domain and range.
To graph the logarithmic function [tex]\( f(x) \)[/tex] with the given points, you can plot the points [tex]\((x,y)\)[/tex] on a suitable coordinate plane. The given points are:
[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]
When you plot these points, you will observe that they lie on a curve typical of a logarithmic function. The characteristic shape of a logarithmic function is such that it increases slowly for small values of [tex]\( x \)[/tex], passes through the point (1, 0), and increases more rapidly as [tex]\( x \)[/tex] grows larger.
### Steps to Plot the Points
1. Plot the points [tex]\((\frac{1}{125}, -3)\)[/tex], [tex]\((\frac{1}{25}, -2)\)[/tex], [tex]\((\frac{1}{5}, -1)\)[/tex], [tex]\((1, 0)\)[/tex], [tex]\((5, 1)\)[/tex], [tex]\((25, 2)\)[/tex], and [tex]\((125, 3)\)[/tex].
2. Use a smooth curve to connect these points, ensuring that the curve passes through each plotted point.
3. Label your axes. Typically, it is useful to label the x-axis with a logarithmic scale in this case for better visualization since x-values span a broad range from [tex]\(\frac{1}{125}\)[/tex] to [tex]\(125\)[/tex].
### Determining the Domain
The logarithmic function [tex]\( f(x) = \log_b(x) \)[/tex] (for any base [tex]\( b \)[/tex]) is only defined for [tex]\( x > 0 \)[/tex]. This is because you cannot take the logarithm of a non-positive number.
Thus, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ x \in (0, \infty) \][/tex]
### Determining the Range
For the given [tex]\( y \)[/tex]-values, we observe that they span from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex]. However, logarithmic functions are not restricted to this specific interval; they can take any real number value given appropriate [tex]\( x \)[/tex]-values.
Therefore, for the general logarithmic function [tex]\( f(x) \)[/tex], the range is:
[tex]\[ y \in (-\infty, \infty) \][/tex]
### Conclusion
In summary:
- Domain of [tex]\( f(x) \)[/tex]: [tex]\( (0, \infty) \)[/tex]
- Range of [tex]\( f(x) \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
By recognizing the nature of logarithmic functions and examining the given data points, you can accurately graph [tex]\( f(x) \)[/tex] and determine its domain and range.
