https://file.aiquickdraw.com/mathbot/file/17268544711615x88xymz.jpg

Let's go through the problem step by step.

The function given is:

\[
f(x) = \frac{|x+1|}{x}
\]

### Part (a): Fill in the table of values.

To compute \( f(x) \) for different values of \( x \), we calculate \( f(x) = \frac{|x+1|}{x} \) for each \( x \) in the table. The student has already attempted filling out the table. Let's verify:

For each \( x \):

1. \( x = -5 \):
   \[
   f(-5) = \frac{|(-5)+1|}{-5} = \frac{|-4|}{-5} = \frac{4}{-5} = -0.8
   \]
   So, \( f(-5) = -0.8 \).

2. \( x = -4 \):
   \[
   f(-4) = \frac{|(-4)+1|}{-4} = \frac{|-3|}{-4} = \frac{3}{-4} = -0.75
   \]
   So, \( f(-4) = -0.75 \).

3. \( x = -3 \):
   \[
   f(-3) = \frac{|(-3)+1|}{-3} = \frac{|-2|}{-3} = \frac{2}{-3} \approx -0.67
   \]
   The student wrote \( -0.66 \), which is an approximation. So, \( f(-3) \approx -0.67 \).

4. \( x = -2 \):
   \[
   f(-2) = \frac{|(-2)+1|}{-2} = \frac{|-1|}{-2} = \frac{1}{-2} = -0.5
   \]
   So, \( f(-2) = -0.5 \).

5. \( x = -1 \):
   \[
   f(-1) = \frac{|(-1)+1|}{-1} = \frac{|0|}{-1} = 0
   \]
   So, \( f(-1) = 0 \).

6. \( x = 1 \):
   \[
   f(1) = \frac{|1+1|}{1} = \frac{2}{1} = 2
   \]
   The student wrote \( 2 \), so that's correct.

7. \( x = 2 \):
   \[
   f(2) = \frac{|2+1|}{2} = \frac{3}{2} = 1.5
   \]
   So, \( f(2) = 1.5 \).

8. \( x = 3 \):
   \[
   f(3) = \frac{|3+1|}{3} = \frac{4}{3} \approx 1.33
   \]
   So, \( f(3) \approx 1.33 \).

9. \( x = 4 \):
   \[
   f(4) = \frac{|4+1|}{4} = \frac{5}{4} = 1.25
   \]
   The student wrote \( 1.25 \), which is correct.

10. \( x = 5 \):
    \[
    f(5) = \frac{|5+1|}{5} = \frac{6}{5} = 1.2
    \]
    So, \( f(5) = 1.2 \).

### Part (b): Are there any values of \( x \) that cannot be inputted into \( f(x) \)?

Yes, \( x = 0 \) cannot be inputted into the function because it would result in division by zero. The expression \( \frac{|x+1|}{x} \) is undefined when \( x = 0 \).

### Part (c): Sketch a graph of \( f(x) \).

The graph of \( f(x) = \frac{|x+1|}{x} \) can be sketched based on the following observations:

- For \( x > 0 \), the function behaves like \( f(x) = \frac{x+1}{x} = 1 + \frac{1}{x} \).
- For \( x < 0 \), the absolute value affects the numerator, giving \( f(x) = \frac{-x-1}{x} = -1 - \frac{1}{x} \).
- There is a vertical asymptote at \( x = 0 \) since the function is undefined there.

To sketch the graph, plot the points from the table and note the behavior of the function approaching \( x = 0 \). The function increases toward \( +\infty \) as \( x \to 0^+ \), and decreases toward \( -\infty \) as \( x \to 0^- \).

Would you like me to draw the graph for you or provide further details?

### Relative Questions:
1. How does the absolute value in the function affect the graph?
2. What is the behavior of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \)?
3. Why does the function have a vertical asymptote at \( x = 0 \)?
4. How would the graph change if there was no absolute value?
5. Can you determine any horizontal asymptotes for this function?

### Tip:
When graphing rational functions, always check for vertical asymptotes (where the denominator is zero) and horizontal asymptotes to guide the shape of the curve.

Consider the function f(x) = |x + 1| / x. (a) Fill in the following table of values. (b) Are there any values of x that cannot be inputted into f? Explain. (c) Sketch a graph of f(x).

Learn how to analyze and graph the function f(x) = |x + 1| / x. This problem covers key algebraic concepts including rational functions, absolute values, and the identification of asymptotes. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation