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.