Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function at each specified value of the independent variable. . \(f(x)=\llbracket-x \rrbracket\) (a) \(f(3)\) (b) \(f(6.1)\) (c) \(f(-5.9)\) (d) \(f(-9)\)

Step-by-Step Solution

Answer

The values of the function \(f(x)\) for the given independent variables are \(f(3) = -3\), \(f(6.1) = -7\), \(f(-5.9) = 5\), and \(f(-9) = 9\).

Step 1: Evaluate f(3)

Plug \(x=3\) into the function, we get \(f(3) = \llbracket -3 \rrbracket = -3\). Since \(-3\) is already an integer, the floor of \(-3\) is \(-3\).

Step 2: Evaluate f(6.1)

For \(x=6.1\), plugging into the function, we obtain \(f(6.1) = \llbracket -6.1 \rrbracket = -7\). Since \(-6.1\) isn't an integer, we find the greatest integer less than \(-6.1\) which is \(-7\).

Step 3: Evaluate f(-5.9)

For \(x=-5.9\), plugging into the function, we get \(f(-5.9) = \llbracket 5.9 \rrbracket = 5\). Despite \(-(-5.9)\) not being an integer, we find the greatest integer less than \(5.9\), which is \(5\).

Step 4: Evaluate f(-9)

Lastly for \(x=-9\), by inserting into the function we find \(f(-9) = \llbracket 9 \rrbracket = 9\). Since \(9\) is an integer, the floor of \(9\) remains \(9\).

Key Concepts

Floor Function

The floor function, also known as the greatest integer function, plays a significant role in mathematics, particularly in discrete mathematics. What does it do? Imagine a number line where you have all the integers marked. When you apply the floor function to a particular number, it gives you the largest integer that is less than or equal to that number. In other words, it 'rounds down' to the nearest whole number.

For instance, if you have a real number like 2.9, the floor function, denoted as \( \lfloor x \rfloor \) where x is our number, will yield 2 since 2 is the greatest integer less than or equal to 2.9. It essentially 'trims off' the decimal part. This is crucial in various applications where results need to be in whole numbers, such as counting objects, indexation in programming, or determining ranges in statistics.

Independent Variable

In the context of evaluating functions, the independent variable is a value you can choose freely. It's the input of the function, often denoted as \(x\). Considering the function \(f(x)\), \(x\) is what we manipulate to get different outputs, or dependent variables, from the function.

Understanding the independent variable is crucial because it represents the variable we have control over or the variable that provides the basis for the function's calculations. When solving problems in mathematics or applying functions in real-world situations, identifying and manipulating the independent variables correctly allows you to predict or determine outcomes.

Function Evaluation

Evaluating a function is akin to running a calculation with a specific input. When we talk about function evaluation, we refer to the process of determining the output of a function for a particular input, the independent variable. The notation \(f(x)\) reads as 'the value of the function \(f\) at \(x\)'.

To evaluate the function, you replace the variable \(x\) with the specified value and simplify the expression following the usual mathematical operations. Remember, every function has its own set of rules, and in the case of the floor function, you'll need to identify the greatest integer less than or equal to the given value as part of the evaluation process.

Greatest Integer Function

The greatest integer function, which is another name for the floor function, is critical when we need to work with integers specifically. In programming, for instance, when we need an index for an array or a list, we often use the greatest integer function to ensure we have a whole number. Similarly, in mathematics, it helps in simplifying complex expressions where whole numbers are needed.

The function is visually represented by a step-like graph, which jumps at every integer value. When evaluating this function for a positive number, we simply 'drop' to the nearest integer below. For negative numbers, we 'climb' to the least integer that is still greater than the given number. This distinction is vital since it affects how we round off decimals and work within number ranges.