Let's analyze the given table and the points provided to graph the function, and then determine its domain and range.
### Step-by-Step Solution:
#### Understanding the Table and Points:
We are given several points [tex]$\left(x, y\right)$[/tex] that represent a relationship between the variables [tex]$x$[/tex] and [tex]$y$[/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]
#### Identifying the Function Type:
Given points suggest a logarithmic relationship because there is a constant rate of change in the [tex]$y$[/tex] values when the [tex]$x$[/tex] values are multiplied or divided by the same factor.
#### Determining the Base of the Logarithm:
From the table:
- When [tex]$x = 1$[/tex], [tex]$y = 0$[/tex], indicating that [tex]$f(1) = \log_b(1) = 0$[/tex], a property true for any logarithmic base [tex]$b$[/tex] since [tex]$\log_b(1) = 0$[/tex].
- When [tex]$x = 5$[/tex], [tex]$y = 1$[/tex], which means [tex]$f(5) = \log_b(5) = 1$[/tex].
This gives us the equation:
[tex]\[ b^1 = 5 \][/tex]
So, the base [tex]$b$[/tex] is [tex]$5$[/tex]. Hence, the function we are dealing with is:
[tex]\[ f(x) = \log_5(x) \][/tex]
#### Graphing the Function:
The function [tex]$\log_5(x)$[/tex] will have the standard logarithmic curve:
- It will pass through [tex]$(1, 0)$[/tex] because [tex]$\log_5(1) = 0$[/tex].
- For [tex]$x > 1$[/tex], [tex]$f(x)$[/tex] increases slowly.
- For [tex]$0 < x < 1$[/tex], [tex]$f(x)$[/tex] decreases and goes towards negative infinity as [tex]$x$[/tex] approaches [tex]$0$[/tex].
#### Determining Domain and Range:
Domain:
- Logarithmic functions are only defined for positive [tex]$x$[/tex] values.
- Therefore, any [tex]$x$[/tex] value must be greater than [tex]$0$[/tex].
- In interval notation: [tex]$(0, \infty)$[/tex].
- In inequality notation: [tex]$x > 0$[/tex].
Range:
- Logarithmic functions can output any real number.
- Thus, the range is all real numbers.
- In interval notation: [tex]$(-\infty, \infty)$[/tex].
- In inequality notation: [tex]$-\infty < y < \infty$[/tex].
#### Final Answer:
- Domain: [tex]$(0, \infty)$[/tex]
- Range: [tex]$(-\infty, \infty)$[/tex]
These answers reflect the nature of logarithmic functions and match the trends observed in the provided table values.
