Answer :
To determine if the relationship represented by the table is a function, we need to check whether each input [tex]\( x \)[/tex] has exactly one output [tex]\( f(x) \)[/tex].
Here's the table you'd need to consider:
[tex]\[
\begin{array}{lr}
x & f(x) \\
51.5 & -46.5 \\
-30.4 & -97.9 \\
-65.7 & 27.4 \\
-92.1 & -47 \\
81.4 & 77.9 \\
\end{array}
\][/tex]
### Steps to Determine if the Relationship is a Function:
1. List the Inputs [tex]\( x \)[/tex]:
Look at the first column of the table. The input values are 51.5, -30.4, -65.7, -92.1, and 81.4.
2. Check for Repeated [tex]\( x \)[/tex]-values:
In a function, each input value [tex]\( x \)[/tex] must map to only one output value [tex]\( f(x) \)[/tex]. If any input [tex]\( x \)[/tex]-value appears more than once, then it must have the same output each time to still be considered a function.
3. Verify Uniqueness:
Check if the [tex]\( x \)[/tex]-values are all unique:
- 51.5
- -30.4
- -65.7
- -92.1
- 81.4
All these values are unique; no value repeats.
4. Conclusion:
Since each [tex]\( x \)[/tex]-value in the table is unique and none of the inputs are repeated, this means each input maps to exactly one output.
Therefore, the relation in the table is indeed a function.
Here's the table you'd need to consider:
[tex]\[
\begin{array}{lr}
x & f(x) \\
51.5 & -46.5 \\
-30.4 & -97.9 \\
-65.7 & 27.4 \\
-92.1 & -47 \\
81.4 & 77.9 \\
\end{array}
\][/tex]
### Steps to Determine if the Relationship is a Function:
1. List the Inputs [tex]\( x \)[/tex]:
Look at the first column of the table. The input values are 51.5, -30.4, -65.7, -92.1, and 81.4.
2. Check for Repeated [tex]\( x \)[/tex]-values:
In a function, each input value [tex]\( x \)[/tex] must map to only one output value [tex]\( f(x) \)[/tex]. If any input [tex]\( x \)[/tex]-value appears more than once, then it must have the same output each time to still be considered a function.
3. Verify Uniqueness:
Check if the [tex]\( x \)[/tex]-values are all unique:
- 51.5
- -30.4
- -65.7
- -92.1
- 81.4
All these values are unique; no value repeats.
4. Conclusion:
Since each [tex]\( x \)[/tex]-value in the table is unique and none of the inputs are repeated, this means each input maps to exactly one output.
Therefore, the relation in the table is indeed a function.
