Problem 6 - Discrete Mathematics with Applications

Question

Let \(f(x)=\lfloor x\rfloor\) and \(g(x)\) \(=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(f \circ g)(-2.3)$$

Step-by-Step Solution

Answer

The value of the composite function \((f \circ g)(-2.3)\) is -2.

Step 1: Evaluate g(-2.3)

To compute the value of the composite function, first evaluate g(-2.3) using the definition of the ceiling function as the smallest integer greater than or equal to the given number. In this case, we have: \[g(-2.3) = \lceil -2.3 \rceil = -2\]

Step 2: Evaluate f(g(-2.3)) = f(-2)

Now that we know the value of g(-2.3), we can evaluate the composite function f(g(-2.3)) using the floor function, which gives the largest integer less than or equal to the given number. In this case, we have \(f(-2)\): \[f(-2) = \lfloor -2 \rfloor = -2\]

Step 3: Final Answer

Thus, the value of the composite function \((f \circ g)(-2.3)\) is -2: \[(f \circ g)(-2.3) = f(g(-2.3)) = f(-2) = -2\]

Key Concepts

Floor Function

The floor function, denoted as \( \lfloor x \rfloor \), is a mathematical operation that rounds a real number down to the nearest integer. It actually gives you the largest integer that is less than or equal to \( x \). For example:

• \( \lfloor 4.7 \rfloor = 4 \)
• \( \lfloor -2.8 \rfloor = -3 \)
• \( \lfloor 3 \rfloor = 3 \) (since 3 is already an integer)

When using the floor function, it's essential to remember that it always "rounds down," meaning it moves towards negative infinity on the number line. This can be a bit counterintuitive with negative numbers since rounding down results in a more negative integer.

Ceiling Function

The ceiling function, represented as \( \lceil x \rceil \), performs the opposite task of the floor function. It rounds a real number up to the nearest integer, giving you the smallest integer that is greater than or equal to \( x \). Here are some examples:

• \( \lceil 4.2 \rceil = 5 \)
• \( \lceil -3.1 \rceil = -3 \)
• \( \lceil 7 \rceil = 7 \) (as 7 is an integer already)

The ceiling function "rounds up," and can be particularly interesting when dealing with negative numbers, as it results in a less negative integer when applied to negative values. This characteristic makes it handy in scenarios where rounding needs to ensure minimum values are not breached.

Real Numbers

Real numbers, symbolized by \( \mathbb{R} \), include every type of number you can think of on the number line. This set consists of:

• Rational numbers: Numbers that can be expressed as fractions like \( \frac{1}{2} \) or repeating decimals like 0.333...
• Irrational numbers: Numbers that cannot be exactly expressed as fractions, like \( \pi \) or \( \sqrt{2} \).
• Integers: Whole numbers including zero, positive numbers, and their negative counterparts such as -5, 0, 23.
• Natural numbers: Positive integers like 1, 2, 3, often used for counting.

Real numbers are fundamental in mathematics because they provide a complete representation of all possible magnitudes. Whether you're measuring length, mass, time, or any other continuous value, real numbers offer the perfect framework for calculation and mathematical modeling.