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Let's analyze the problem.

### Given Function:
\[ f(x) = \frac{x+1}{x} \]

### (a) Fill in the table:
The values of \( y \) for corresponding \( x \) are calculated as follows:
\[
f(x) = \frac{x+1}{x} = 1 + \frac{1}{x}
\]

- For \( x = -5 \), \( f(x) = 1 + \frac{1}{-5} = 1 - 0.2 = 0.8 \)
- For \( x = -4 \), \( f(x) = 1 + \frac{1}{-4} = 1 - 0.25 = 0.75 \)
- For \( x = -3 \), \( f(x) = 1 + \frac{1}{-3} = 1 - 0.33 \approx 0.67 \)
- For \( x = -2 \), \( f(x) = 1 + \frac{1}{-2} = 1 - 0.5 = 0.5 \)
- For \( x = -1 \), \( f(x) = 1 + \frac{1}{-1} = 1 - 1 = 0 \)
- For \( x = 1 \), \( f(x) = 1 + \frac{1}{1} = 1 + 1 = 2 \)
- For \( x = 2 \), \( f(x) = 1 + \frac{1}{2} = 1 + 0.5 = 1.5 \)
- For \( x = 3 \), \( f(x) = 1 + \frac{1}{3} = 1 + 0.33 \approx 1.33 \)
- For \( x = 4 \), \( f(x) = 1 + \frac{1}{4} = 1 + 0.25 = 1.25 \)
- For \( x = 5 \), \( f(x) = 1 + \frac{1}{5} = 1 + 0.2 = 1.2 \)

The values seem mostly correct, but for \( x = -3 \), the calculated value is approximately \( 0.67 \), not \( 0.5 \), and for \( x = -4 \), it should be \( 0.75 \) instead of \( 0.86 \).

### (b) Are there any values of \( x \) that cannot be inputted into \( f \)?
Yes, \( x = 0 \) cannot be inputted because it would result in division by zero, which is undefined. The function \( f(x) \) includes \( \frac{1}{x} \), and division by zero is not allowed.

### (c) Sketch of the graph of \( f(x) \):
- As \( x \to 0 \), the function approaches a vertical asymptote since \( f(x) \) tends toward \( \infty \) for \( x > 0 \) and \( -\infty \) for \( x < 0 \).
- For large positive and negative values of \( x \), the function approaches 1, creating a horizontal asymptote at \( y = 1 \).

I can help you sketch the graph based on these observations. Would you like me to generate the graph?

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Do you want any further clarification or details?

Here are five related questions for deeper understanding:
1. What happens to the function \( f(x) \) as \( x \) approaches 0 from the positive side?
2. What is the behavior of \( f(x) \) as \( x \to \infty \)?
3. How do vertical and horizontal asymptotes affect the shape of a graph?
4. Can you identify points where the function is increasing or decreasing?
5. How would you describe the domain and range of \( f(x) \)?

Tip: Asymptotes play a key role in understanding the behavior of rational functions like \( f(x) \), especially at extreme values of \( x \).

Consider the function f(x) = (x + 1) / x. (a) Fill in the table of values for x between -5 and 5. (b) Are there any values of x that cannot be inputted into f? (c) Sketch a graph of f(x).

Explore the function f(x) = (x + 1) / x, including a table of values, identifying undefined points, and graphing the function with asymptotic behavior. Learn how division by zero impacts the function's domain and the presence of asymptotes.

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