Problem 79 - College Algebra

Question

In Exercises, if \(f(x)=\) int \((x),\) find each function value. $$f\left(\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The value of the function at \(f\left(\frac{1}{3}\right)\) is zero.

Step 1: Understand the Function

The function is defined as \(f(x) = \) int \((x)\). This function returns the integer part of x, removing any decimal places.

Step 2: Substitute the Value to the Function

Substitute \(\frac{1}{3}\) into the function, where x is equal to \(\frac{1}{3}\), we get: \(f\left(\frac{1}{3}\right) = \) int\(\left(\frac{1}{3}\right)\)

Step 3: Finding the Integer Part

The integer part of \(\frac{1}{3}\) is zero, because \(\frac{1}{3}\) is less than 1, thus it does not have any whole number part. This gives us, \(f\left(\frac{1}{3}\right) = 0\)

Key Concepts

Floor Function

The floor function, often denoted as \( \text{int}(x) \) or \( \lfloor x \rfloor \), plays a fundamental role in mathematics, especially in areas dealing with integers. It can be understood as the greatest integer less than or equal to a given number. In other words, when you apply the floor function to a number, you round it down to the nearest whole number.

For example, the floor of 2.9 is 2, and the floor of -2.4 is -3. It's important to remember that it always rounds towards negative infinity, meaning for positive numbers it rounds down, and for negative numbers, it rounds even more negative.

Here is how you can easily visualize it:

• The floor of a positive decimal like 3.4 is 3.
• The floor of a negative decimal like -1.2 becomes more negative, resulting in -2.

Understanding this concept is crucial for computations in both mathematical theory and computer algorithms.

Function Evaluation

Evaluating a function means finding the value of the function at a particular input, which is a core activity in algebra. Given a function \( f(x) \), and a value to substitute for \( x \), the goal is simply to compute \( f(x) \).In our exercise, the function is \( f(x) = \text{int}(x) \), and the task is to evaluate this function at \( x = \frac{1}{3} \). To do this, we substitute \( \frac{1}{3} \) into the function equation. The floor function, \( \text{int}(x) \), returns the largest integer less than or equal to the input value. Thus, \( f\left(\frac{1}{3}\right) = 0 \) since zero is the greatest integer less than \( \frac{1}{3} \).Here are some steps to help with function evaluation for similar purposes:

• Identify and understand the function and its rules.
• Substitute the given value into the function.
• Apply the specific function rule, such as the floor function, to compute the result.

Function evaluation requires practice, helping you understand both the process and the result.

Algebra Exercises

Algebra exercises like this one help reinforce foundational concepts such as functions, substitutions, and operations on numbers. They are designed to help students understand how different mathematical tools work together.

In exercises involving functions, particularly the integer or floor function, it's vital to follow a systematic process:

• Start by understanding the definitions and properties of the function.
• Substitute any provided values directly into the function.
• Apply the function's rule to find the result.

Keeping this process in mind will not only help solve the current problem but will also prepare you for more complex algebraic tasks. Grasping these underlying concepts enables a better understanding of algebra's logic, setting a solid foundation for more advanced mathematics.