To graph the logarithmic function [tex]\(f(x)\)[/tex] and determine its domain and range, let's analyze the given table and understand the properties of logarithmic functions in general.
### Understanding the Table
The table shows pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values for the logarithmic 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]
### Domain of [tex]\(f(x)\)[/tex]
For any logarithmic function [tex]\(f(x) = \log_b(x)\)[/tex] where [tex]\(b > 1\)[/tex] (the base of the logarithm), the domain is the set of all positive real numbers, since the logarithm is only defined for positive values of [tex]\(x\)[/tex].
From the table:
- There's a correspondence for positive [tex]\(x\)[/tex] values ranging from [tex]\(\frac{1}{125}\)[/tex] to [tex]\(125\)[/tex].
Thus, the domain of [tex]\(f(x)\)[/tex] is:
[tex]\[
(0, \infty)
\][/tex]
### Range of [tex]\(f(x)\)[/tex]
The range of a logarithmic function [tex]\(f(x) = \log_b(x)\)[/tex] is the set of all real numbers, since as [tex]\(x\)[/tex] increases, [tex]\(f(x)\)[/tex] increases without bound, and as [tex]\(x\)[/tex] approaches zero from the positive side, [tex]\(f(x)\)[/tex] decreases without bound.
From the table:
- [tex]\(y\)[/tex] values span from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex], representing a small section of the overall range which extends infinitely in both directions.
Thus, the range of [tex]\(f(x)\)[/tex] is:
[tex]\[
(-\infty, \infty)
\][/tex]
### Plotting the Function
Using the given pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values from the table, we can plot points on a coordinate plane:
[tex]\[
\begin{align*}
(\frac{1}{125}, -3), & \quad (\frac{1}{25}, -2), \\
(\frac{1}{5}, -1), & \quad (1, 0), \\
(5, 1), & \quad (25, 2), \\
(125, 3)
\end{align*}
\][/tex]
These points can be connected smoothly because we know logarithmic functions have a continuous curve. The graph will show an asymptotic behavior towards the y-axis (as [tex]\(x\)[/tex] approaches 0 from the positive side), passing through (1, 0), and increasing without bound.
### Final Answer
Domain:
[tex]\[
(0, \infty)
\][/tex]
Range:
[tex]\[
(-\infty, \infty)
\][/tex]
Explanation:
The domain of the logarithmic function [tex]\(f(x)\)[/tex] includes all positive real numbers because [tex]\( \log_b(x) \)[/tex] is defined for [tex]\(x > 0\)[/tex]. The range includes all real numbers because logarithmic functions can take on any real value as [tex]\(x\)[/tex] varies over the positive real numbers.
By plotting the points from the given table and understanding the properties of logarithmic functions, you can sketch a continuous curve that asymptotically approaches the y-axis and increases indefinitely, confirming the stated domain and range.
