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To determine the correct answer for this question, let’s analyze each function and option.

1. **Given Functions**:
   - The linear function \( f(x) = x + 1 \): This is a straight line with a slope of 1 and y-intercept of 1.
   - The absolute value function \( f(x) = |x| - 1 \): This creates a V-shaped graph, with its vertex at \( (0, -1) \).

2. **Domain and Range of Each Function**:
   - **Domain**:
     - For both \( f(x) = x + 1 \) and \( f(x) = |x| - 1 \), the domain is all real numbers, \( (-\infty, \infty) \), because both functions accept any real number as input.
   - **Range**:
     - For \( f(x) = x + 1 \), the range is also all real numbers \( (-\infty, \infty) \), as a linear function has no vertical restrictions.
     - For \( f(x) = |x| - 1 \), the range is \( [-1, \infty) \), since the absolute value function ensures that \( f(x) \geq -1 \) for all \( x \).

3. **Analysis of Answer Choices**:
   - **Option A**: "The range of both functions is \( (-\infty, \infty) \)."  
     - This is incorrect because the range of \( f(x) = |x| - 1 \) does not include values below \(-1\).
   - **Option B**: "The domain of both functions is \( (-\infty, \infty) \)."  
     - This is correct, as both functions can take any real number as input.
   - **Option C**: "The domain of both functions is \( [1, \infty) \)."  
     - This is incorrect because the domain of both functions is not restricted to positive values; they accept all real numbers.
   - **Option D**: "The range of both functions is \( [1, \infty) \)."  
     - This is incorrect because the range of \( f(x) = |x| - 1 \) starts from \(-1\), not 1.

4. **Conclusion**:
   - The correct answer is **Option B**, "The domain of both functions is \( (-\infty, \infty) \)".

Would you like further details or have any questions?

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**Follow-Up Questions:**

1. What is the effect of changing the slope in a linear function?
2. How would the range of \( f(x) = |x| - 1 \) change if the -1 were replaced with another constant?
3. What would be the range of \( f(x) = |x + 1| \)?
4. How do transformations affect the vertex of an absolute value function?
5. How does the graph of \( f(x) = |x| \) differ from \( f(x) = |x| - 1 \)?

**Tip:** When working with absolute value functions, pay attention to the vertex and how transformations (like shifts) impact it.

The graph shows two functions: a linear function f(x) = x + 1 and an absolute value function f(x) = |x| - 1. Choose the correct statement about their domain and range.

Explore the domain and range of the linear function f(x) = x + 1 and the absolute value function f(x) = |x| - 1. Learn why both functions have a domain of (-∞, ∞) and analyze the differences in their ranges. This problem is perfect for Grades 8-10 students learning about functions.

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