Domain Of Step Function F(x) = Ceil(2x) - 1

When we talk about the domain of a function, we're essentially asking: "What kind of input values can we legally plug into this function?" For the step function $f(x) = \lceil 2x\rceil - 1$, we need to carefully consider the operations involved to determine the set of all possible $x$ values. The ceiling function, denoted by $\lceil y\rceil$, gives us the smallest integer greater than or equal to $y$. In our function, the input to the ceiling function is $2x$. Since $x$ can be any real number, $2x$ can also be any real number. The ceiling function is defined for all real numbers. Therefore, no matter what real number you choose for $x$, $2x$ will be a real number, and the ceiling function $\lceil 2x\rceil$ will produce a well-defined integer. Subsequently, subtracting 1 from this integer will also result in a well-defined integer. This means that the expression $\lceil 2x\rceil - 1$ is defined for all real numbers $x$. Thus, the domain of the function $f(x) = \lceil 2x\rceil - 1$ is the set of all real numbers.

Let's delve a bit deeper into why the domain is the set of all real numbers. The core of our function is the ceiling operation, $\lceil \cdot \rceil$. This mathematical tool takes any real number as its input and outputs the smallest integer that is greater than or equal to that input. For instance, $\lceil 3.14 \rceil = 4$, $\lceil -2.7 \rceil = -2$, and $\lceil 5 \rceil = 5$. The key here is that any real number can be fed into the ceiling function, and it will always yield a sensible integer result. Now, consider the argument of our ceiling function: $2x$. If $x$ can be any real number, then $2x$ can also be any real number. There are no restrictions on what values $2x$ can take if $x$ is allowed to be any real number. For example, if $x = 0.5$, then $2x = 1$, and $\lceil 1 \rceil = 1$. If $x = \pi$, then $2x = 2\pi \approx 6.283$, and $\lceil 2\pi \rceil = 7$. If $x = -100.9$, then $2x = -201.8$, and $\lceil -201.8 \rceil = -201$. In every scenario, the operation $\lceil 2x\rceil$ is perfectly valid.

After the ceiling function does its job, we subtract 1: $f(x) = \lceil 2x\rceil - 1$. Since $\lceil 2x\rceil$ always produces an integer, subtracting 1 from an integer will always result in another integer. For example, if $\lceil 2x\rceil = 5$, then $f(x) = 5 - 1 = 4$. If $\lceil 2x\rceil = -3$, then $f(x) = -3 - 1 = -4$. There's no division by zero to worry about, no square roots of negative numbers, and no logarithms of non-positive numbers. All the operations are safe and well-defined for any real number $x$ that we might input. Therefore, the set of all possible values for $x$ that we can use in this function is the entire set of real numbers. This is why the domain is denoted as ${x \mid x \text{ is a real number }}$. This is a fundamental concept when analyzing functions, and understanding the domain is the first step in a comprehensive analysis.

Understanding the Options

Let's examine why the other options are not correct for the domain of $f(x) = \lceil 2x\rceil - 1$. Option A suggests the domain is ${x \mid x \geq -1}$. This implies that any real number less than -1, such as -1.5 or -2, cannot be used as an input. However, as we've established, we can plug in any real number. For instance, if $x = -2$, then $2x = -4$, and $\lceil -4 \rceil - 1 = -4 - 1 = -5$. This is a valid output, so $x = -2$ should be in the domain. Similarly, if $x = -1.5$, $2x = -3$, and $\lceil -3 \rceil - 1 = -3 - 1 = -4$, which is also a valid output. Therefore, the restriction $x \geq -1$ is incorrect.

Option B proposes the domain as ${x \mid x \geq 1}$. This is even more restrictive than Option A and is also incorrect for the same reasons. It excludes valid inputs like $x = 0$, $x = 0.5$, or $x = -3$. If we try $x = 0$, then $2x = 0$, and $\lceil 0 \rceil - 1 = 0 - 1 = -1$. This is a perfectly valid computation. If we try $x = 0.5$, we get $f(0.5) = \lceil 2 \times 0.5 \rceil - 1 = \lceil 1 \rceil - 1 = 1 - 1 = 0$. If we try $x = -3$, $f(-3) = \lceil 2 \times (-3) \rceil - 1 = \lceil -6 \rceil - 1 = -6 - 1 = -7$. All these examples show that values of $x$ less than 1 are valid inputs.

Option C suggests that the domain is ${x \mid x \text{ is an integer }}$. This means we can only input whole numbers (positive, negative, or zero). However, the definition of the ceiling function works for all real numbers, not just integers. If we restrict $x$ to be an integer, we are unnecessarily limiting the function's applicability. For example, $x = 0.25$ is a real number. If we input it into our function, we get $f(0.25) = \lceil 2 \times 0.25 \rceil - 1 = \lceil 0.5 \rceil - 1 = 1 - 1 = 0$. This is a valid output, but $x = 0.25$ is not an integer, so this option would incorrectly exclude it from the domain.

The Correct Domain: All Real Numbers

As we've thoroughly explored, the operations within the function $f(x) = \lceil 2x\rceil - 1$ do not impose any restrictions on the value of $x$. The ceiling function $\lceil y\rceil$ is defined for all real numbers $y$. Since the argument of our ceiling function is $2x$, and $2x$ can represent any real number when $x$ is any real number, the ceiling function is always well-defined. Subtracting 1 from an integer always yields an integer. Therefore, the function is defined for every single real number. The set of all real numbers is typically denoted by the symbol $\mathbb{R}$ or by the set-builder notation ${x \mid x \text{ is a real number }}$. This means you can pick any number from the number line, positive, negative, zero, fractions, decimals, irrational numbers – they all work as inputs for this step function.

Visualizing the Step Function

While understanding the domain is about what you can input, it's also helpful to visualize what the function does. The function $f(x) = \lceil 2x\rceil - 1$ is a type of step function. Step functions are characterized by having constant values over intervals, and then jumping to a new value at specific points. For $f(x) = \lceil 2x\rceil - 1$, the jumps occur when $2x$ becomes an integer. This happens when $x$ is a half-integer (e.g., $0.5, 1.5, -2.5$). For example:

• If $0 \leq x < 0.5$, then $0 \leq 2x < 1$, so $\lceil 2x\rceil = 1$, and $f(x) = 1 - 1 = 0$.
• If $0.5 \leq x < 1$, then $1 \leq 2x < 2$, so $\lceil 2x\rceil = 2$, and $f(x) = 2 - 1 = 1$.
• If $1 \leq x < 1.5$, then $2 \leq 2x < 3$, so $\lceil 2x\rceil = 3$, and $f(x) = 3 - 1 = 2$.
• If $-0.5 \leq x < 0$, then $-1 \leq 2x < 0$, so $\lceil 2x\rceil = 0$, and $f(x) = 0 - 1 = -1$.

This pattern continues for all real numbers. The graph of this function would consist of horizontal line segments at different integer $y$-values, with jumps occurring at specific $x$-values. The crucial takeaway for the domain, however, remains that any real number $x$ can be used to generate these steps and values.

Conclusion

In summary, when determining the domain of a function, we look for any values of the input variable that would lead to an undefined mathematical operation. For $f(x) = \lceil 2x\rceil - 1$, the ceiling function $\lceil 2x\rceil$ is defined for all real numbers $x$, and subtracting 1 from the result is also always defined. Therefore, there are no restrictions on $x$. The correct domain for this step function is the set of all real numbers.

For further exploration into functions and their domains, you can visit Khan Academy's Mathematics section for comprehensive resources and tutorials.