Problem 8 - Discrete Mathematics with Applications

Question

Let \(f(x)=\lfloor x\rfloor\) and \( g(x)=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(f \circ g)(-3.9)$$

Step-by-Step Solution

Answer

\((f \circ g)(-3.9) = -3\)

Step 1: Evaluate the inner function g(x)

First, we need to compute the inner function, i.e., \(g(-3.9)\). We will find the smallest integer greater than or equal to -3.9. In this case, the smallest integer greater than or equal to -3.9 is -3. Therefore, we get \(g(-3.9) = -3\).

Step 2: Evaluate the composition f(g(x))

Now that we have found the value of the inner function, we need to substitute this value into the outer function f(x). So, we will compute the greatest integer less than or equal to -3. Since -3 is already an integer, the greatest integer less than or equal to -3 is -3 itself. Therefore, we get \(f(g(-3.9)) = f(-3) = -3\). So, \((f \circ g)(-3.9) = -3\).

Key Concepts

Function Composition

Function composition is a mathematical operation where two functions, say \( f \) and \( g \), are combined to form a third function often denoted as \( f(g(x)) \). This essentially means applying one function to the results of another.
This can be visualized as a two-step process:

• The first step involves taking an input \( x \) and putting it through the function \( g \).
• Then, take the result of \( g(x) \) and use it as the input for the function \( f \), thereby calculating \( f(g(x)) \).

Function composition is typically written as \( (f \circ g)(x) = f(g(x)) \).
The order of composition is important since \( f(g(x)) \) may not yield the same outcome as \( g(f(x)) \). Understanding function composition helps solve problems where input values undergo multiple transformations before reaching the final result.

Real Numbers

Real numbers, denoted by \( \mathbb{R} \), comprise all the possible numbers you can think of, except for complex numbers. This includes:

• Whole numbers like \( 0, 1, 2, \) etc.
• Integers such as \(..., -2, -1, 0, 1, 2, ...\).
• Rational numbers which can be expressed as the quotient of two integers, like \( \frac{1}{2}, \frac{3}{4}, -\frac{5}{3} \).
• Irrational numbers which cannot be written as a simple fraction, like \( \sqrt{2} \) or \( \pi \).

Real numbers form a continuous line of numbers without any gaps, covering everything from negative infinity to positive infinity.
In the context of floor and ceiling functions, real numbers are important because the functions map any real number to the nearest integer in specific ways, either by rounding down (floor) or rounding up (ceiling). Understanding the real numbers allows you to comprehend how these functions transform the inputs mathematically.

Integer Functions

Integer functions like floor and ceiling functions play a crucial role in rounding real numbers to integers. Let's explore these two functions:

• Floor Function: The floor function, denoted as \( \lfloor x \rfloor \), maps a real number \( x \) to the greatest integer less than or equal to \( x \). It effectively rounds down the number. For example, \( \lfloor 3.9 \rfloor = 3 \) and \( \lfloor -2.2 \rfloor = -3 \).
• Ceiling Function: Conversely, the ceiling function, denoted as \( \lceil x \rceil \), maps a real number \( x \) to the smallest integer greater than or equal to \( x \). This means it rounds the number up. For example, \( \lceil 2.3 \rceil = 3 \) and \( \lceil -4.1 \rceil = -4 \).

Both functions help in simplifying expressions involving real numbers, as they convert continuous values into discrete integers.
By understanding how these integer functions operate, particularly in function compositions, students can solve complex problems like the one presented in the exercise. The floor function \( f(x) = \lfloor x \rfloor \) rounds down to the nearest integer, while the ceiling function \( g(x) = \lceil x \rceil \) always rounds up.