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 1 Over x minus 2 EndFraction
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Part 1
x
minus2
minus1
0
1
2
3
4
​f(x)equalsStartFraction 1 Over x minus 2 EndFraction
  
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​(Type integers or simplified fractions. Type N if the function is​ undefined.)

Answer 1

To compute the values of the function \( f(x) = \frac{1}{x - 2} \) for the given x-values, we will evaluate the function for each \( x \).

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

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

3. **For \( x = 0 \)**:
\( f(0) = \frac{1}{0 - 2} = \frac{1}{-2} = -\frac{1}{2} \)

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

5. **For \( x = 2 \)**:
\( f(2) = \frac{1}{2 - 2} \) is undefined. So, we will write **N** (not defined).

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

7. **For \( x = 4 \)**:
\( f(4) = \frac{1}{4 - 2} = \frac{1}{2} \)

Now we can summarize the table of values:

| \( x \) | \( f(x) \) |
|---------|-------------------------|
| -2 | -\(\frac{1}{4}\) |
| -1 | -\(\frac{1}{3}\) |
| 0 | -\(\frac{1}{2}\) |
| 1 | -1 |
| 2 | N |
| 3 | 1 |
| 4 | \(\frac{1}{2}\) |

### Summary of Results:

- \( f(-2) = -\frac{1}{4} \)
- \( f(-1) = -\frac{1}{3} \)
- \( f(0) = -\frac{1}{2} \)
- \( f(1) = -1 \)
- \( f(2) = N \) (undefined)
- \( f(3) = 1 \)
- \( f(4) = \frac{1}{2} \)

### Graphing the Points:
On a coordinate system, you can plot the points:
- (-2, -1/4)
- (-1, -1/3)
- (0, -1/2)
- (1, -1)
- (3, 1)
- (4, 1/2)

### Note on the function:
The function has a vertical asymptote at \( x = 2 \), which means that the function approaches infinity or negative infinity as \( x \) approaches 2 from the left or right. Thus, in a graph, there will be a break at that point.

You can sketch the graph based on these points and the characteristics of the function!