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To determine the domain and range of the function \( f(x) = |x| + 4 \), let's analyze it:

1. **Domain**: The absolute value function \( |x| \) is defined for all real values of \( x \). Adding 4 does not restrict the values of \( x \), so the domain is **all real numbers**.

2. **Range**: The function \( f(x) = |x| + 4 \) takes the absolute value of \( x \) and adds 4. Since \( |x| \geq 0 \), the smallest value of \( f(x) \) is when \( x = 0 \), giving \( f(0) = 0 + 4 = 4 \). Therefore, \( f(x) \geq 4 \), so the range is **\( y \geq 4 \)**.

Now, let’s match these with the options provided:

- **Option (a)**: Domain is all real numbers, Range \( y \leq 4 \) – Incorrect (Range should be \( y \geq 4 \))
- **Option (b)**: Domain is all real numbers, Range \( y \geq 4 \) – Correct
- **Option (c)**: Domain is all real numbers, Range \( y < 4 \) – Incorrect (Range should include values equal to and greater than 4)
- **Option (d)**: Domain \( x \geq 4 \), Range is all real numbers – Incorrect (Domain is all real numbers, not restricted to \( x \geq 4 \))

**Correct Answer:** **Option (b)** – Domain: all real numbers, Range: \( y \geq 4 \).

Would you like further details or have questions about this solution?

Here are some related questions to explore further:
1. What is the impact on the range if we modify the function to \( f(x) = |x| - 4 \)?
2. How would the graph look if we used \( f(x) = -|x| + 4 \)?
3. How do we determine the domain and range for functions involving square roots?
4. What happens to the range if we add or subtract a constant from an absolute value function?
5. How can transformations affect the graph of absolute value functions?

**Tip:** For absolute value functions of the form \( f(x) = |x| + c \), the range starts at \( c \) (if \( c \geq 0 \)) and extends to infinity.

What is the domain and range of the absolute value function f(x) = |x| + 4?

Determine the domain and range of the absolute value function f(x) = |x| + 4 with a detailed analysis. Explore key concepts in algebra, including absolute value properties and how they impact the range. Suitable for students in Grades 9-10.

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