To find the range of the function $f(x) = |3 - x| + |2 + x|$ over the interval $x \in [0, 4]$, we need to consider how the absolute value terms behave. Let's break it down into pieces.

1. Behavior of $|3 - x|$ and $|2 + x|$:

• For $|3 - x|$, it will behave differently depending on whether $x$ is less than 3 or greater than 3.
• For $|2 + x|$, it will behave differently depending on whether $x$ is less than -2 or greater than -2.

In the given interval $[0, 4]$, both expressions inside the absolute value functions are always positive. Therefore, we can remove the absolute values as follows:

• $|3 - x| = 3 - x$ for $x \leq 3$, and $|3 - x| = x - 3$ for $x > 3$
• $|2 + x| = 2 + x$ for all $x \geq 0$

2. Piecewise Function:

Now, we will split the interval $[0, 4]$ into parts where the expressions inside the absolute values change:

• For $0 \leq x \leq 3$: $f(x) = (3 - x) + (2 + x) = 5$

• For $3 < x \leq 4$: $f(x) = (x - 3) + (2 + x) = 2x - 1$

3. Find Range:

Now, let's evaluate the function at key points in the interval:

• At $x = 0$, $f(0) = 5$
• At $x = 3$, $f(3) = 5$
• At $x = 4$, $f(4) = 2(4) - 1 = 7$

Thus, $f(x)$ takes values from 5 to 7 over the interval $[0, 4]$.

4. Conclusion:

The range of $f(x)$ over $x \in [0, 4]$ is $[5, 7]$.

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Would you like more details about how the piecewise function was derived? Here are some related questions:

1. How do you solve absolute value functions in piecewise form?
2. What happens when the function involves both positive and negative values inside the absolute value?
3. How can you determine the points where a function changes behavior (like where the absolute values change)?
4. What are some other ways to find the range of a function over a given interval?
5. How would you analyze a function like this over a different interval (e.g., $[-2, 2]$)?

Tip: When dealing with absolute value functions, always identify the points where the expressions inside the absolute value functions change sign, as they can change the way the function behaves!