Question

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Part 1
Complete the table of values for the following function and then plot the points to graph the function by hand.
​f(x)equalsStartFraction 3 Over x minus 2 EndFraction
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Part 1
x
minus2
minus1
0
1
2
3
4
​f(x)equalsStartFraction 3 Over x minus 2 EndFraction
  
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enter your response here
  
enter your response here
  
enter your response here
  
enter your response here
  
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​(Type integers or simplified fractions. Type N if the function is​ undefined.)

Answer 1

To find the values of the function \( f(x) = \frac{3}{x - 2} \), we will evaluate this function for each value of \( x \) in the provided table: \( -2, -1, 0, 1, 2, 3, 4 \).

Let's calculate \( f(x) \) for each \( x \):

1. For \( x = -2 \):
\[ f(-2) = \frac{3}{-2 - 2} = \frac{3}{-4} = -\frac{3}{4} \]

2. For \( x = -1 \):
\[ f(-1) = \frac{3}{-1 - 2} = \frac{3}{-3} = -1 \]

3. For \( x = 0 \):
\[ f(0) = \frac{3}{0 - 2} = \frac{3}{-2} = -\frac{3}{2} \]

4. For \( x = 1 \):
\[ f(1) = \frac{3}{1 - 2} = \frac{3}{-1} = -3 \]

5. For \( x = 2 \):
\[ f(2) = \frac{3}{2 - 2} = \frac{3}{0} \rightarrow \text{undefined} \]

6. For \( x = 3 \):
\[ f(3) = \frac{3}{3 - 2} = \frac{3}{1} = 3 \]

7. For \( x = 4 \):
\[ f(4) = \frac{3}{4 - 2} = \frac{3}{2} \]

Now, let's compile these results into the table:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -\frac{3}{4} \\
-1 & -1 \\
0 & -\frac{3}{2} \\
1 & -3 \\
2 & \text{N (undefined)} \\
3 & 3 \\
4 & \frac{3}{2} \\
\hline
\end{array}
\]

### Summary of Values:
- \( f(-2) = -\frac{3}{4} \)
- \( f(-1) = -1 \)
- \( f(0) = -\frac{3}{2} \)
- \( f(1) = -3 \)
- \( f(2) = \text{undefined} \)
- \( f(3) = 3 \)
- \( f(4) = \frac{3}{2} \)

### Points to Plot:
- \( (-2, -\frac{3}{4}) \)
- \( (-1, -1) \)
- \( (0, -\frac{3}{2}) \)
- \( (1, -3) \)
- \( (2, \text{undefined}) \)
- \( (3, 3) \)
- \( (4, \frac{3}{2}) \)

To graph the function by hand, plot each of the calculated points and remember that as \( x \) approaches 2, \( f(x) \) approaches \( \pm \infty \), creating a vertical asymptote at \( x = 2 \).