To determine the domain and range of the function described by the table, let's follow these steps:
1. Understand the Table: The table provided lists pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values. Here are those values in a more readable format:
[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]
2. List the Domain: The domain of a function consists of all the input [tex]\( x \)[/tex]-values. Given the pairs from the table, we identify these [tex]\( x \)[/tex]-values:
[tex]\[
x = \left\{ \frac{1}{125}, \frac{1}{25}, \frac{1}{5}, 1, 5, 25, 125 \right\}
\][/tex]
To make it more readable, we can convert those fractions to decimals where appropriate:
[tex]\[
x = \left\{ \frac{1}{125} = 0.008, \frac{1}{25} = 0.04, \frac{1}{5} = 0.2, 1, 5, 25, 125 \right\}
\][/tex]
Therefore, the domain is:
[tex]\[
\text{Domain} = \{0.008, 0.04, 0.2, 1, 5, 25, 125\}
\][/tex]
3. List the Range: The range of a function consists of all the output [tex]\( y \)[/tex]-values. The [tex]\( y \)[/tex]-values given in the table are:
[tex]\[
y = \{-3, -2, -1, 0, 1, 2, 3\}
\][/tex]
Thus, the range of the function is:
[tex]\[
\text{Range} = \{-3, -2, -1, 0, 1, 2, 3\}
\][/tex]
4. Graph the Function: To graph the function, you would plot each [tex]\( x \)[/tex]-value against its corresponding [tex]\( y \)[/tex]-value on a coordinate plane. The pairs you'll plot are:
[tex]\[
(0.008, -3), (0.04, -2), (0.2, -1), (1, 0), (5, 1), (25, 2), (125, 3)
\][/tex]
When plotting, you mark each of these ordered pairs on the graph, ensuring to accurately represent both the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-axes values.
In Summary:
- Domain: \{0.008, 0.04, 0.2, 1, 5, 25, 125\}
- Range: \{-3, -2, -1, 0, 1, 2, 3\}
These are the set of all possible input values (domain) and all possible output values (range) for the function.
