All points of the step function *f*(*x*) are graphed.

What is the domain of *f*(*x*)?

{*x*| –4 < *x* ≤ 4}

{*x*| –3 < *x* ≤ 4}

{*x*| 1 < *x* ≤ 4}

{*x*| 2 < *x* ≤ 4}

If all points of the step function \( f(x) = \lceil 2x \rceil - 1 \) are graphed, then the domain of \( f(x) \) would be the set of \( x \) values for which the function is defined and included in the graph.

Given the choices, it suggests that the graph is only considering a limited range of \( x \) values.

Let's analyze the options:

1. **{*x* | –4 < *x* ≤ 4}**  
   This includes all real numbers \( x \) within the interval from just greater than \(-4\) to \(4\). The function would be defined and graphed for all \( x \) in this interval.

2. **{*x* | –3 < *x* ≤ 4}**  
   This includes all real numbers \( x \) within the interval from just greater than \(-3\) to \(4\).

3. **{*x* | 1 < *x* ≤ 4}**  
   This includes all real numbers \( x \) within the interval from just greater than \(1\) to \(4\).

4. **{*x* | 2 < *x* ≤ 4}**  
   This includes all real numbers \( x \) within the interval from just greater than \(2\) to \(4\).

Since the graph must show all points of the function within a given interval, the correct choice should be the interval that represents the full range where the function is graphed.

The most comprehensive range provided in the choices is:

**{*x* | –4 < *x* ≤ 4}**

This would include all the possible values of \( x \) that could be part of the graph over that interval.

Would you like to explore more about this, or do you have any questions?

Here are some related questions:

1. How does the step size of the function \( f(x) = \lceil 2x \rceil - 1 \) affect the graph?
2. What would be the range of \( f(x) \) if the domain were \( –4 < x \leq 4 \)?
3. How would the graph change if the domain was \( 1 < x \leq 4 \)?
4. What is the effect of the interval \( x \) on the continuity of the step function?
5. How can the function \( f(x) = \lceil 2x \rceil - 1 \) be defined piecewise?

**Tip:** When determining the domain from a graph, always consider the interval of \( x \)-values for which the graph exists, and ensure that all points within that interval are covered.

Explore the domain of the step function f(x) = ceil(2x) - 1 through detailed analysis. Learn about how the domain influences the graph and which x-values are included in the function's graph. Suitable for high school students studying functions and their domains.

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