Problem 78 - Beginning and Intermediate Algebra

Question

Let \(f(x)=[x] .\) Find the following function values. $$f\left(-1 \frac{3}{4}\right)$$

Step-by-Step Solution

Answer

The function value is \(f\left(-1\frac{3}{4}\right) = -2\).

Step 1: Understand the meaning of the greatest integer function

The greatest integer function, denoted as \([x]\), is the largest integer that is less than or equal to the input value \(x\). We will use this definition to determine the value of the function at the given point.

Step 2: Evaluate the function at the given point

Now we will evaluate the function at the given point \(x = -1\frac{3}{4}\). We need to find the largest integer that is less than or equal to \(-1\frac{3}{4}\). Since \(-1\frac{3}{4}\) is between \(-2\) and \(-1\), the largest integer less than or equal to \(-1\frac{3}{4}\) is \(-2\). Therefore, the function value at the given point is: \[f\left(-1\frac{3}{4}\right) = [-1\frac{3}{4}] = -2\]

Key Concepts

Step by Step Solution

In solving problems like the one above, it's helpful to break the process down into clear, manageable steps. This can help ensure you understand each part of the solution. Let's take the greatest integer function and walk through how to solve it step by step.

To start, it's important to clearly understand what the greatest integer function, denoted \(\[x\]\), means. It represents the largest integer that is less than or equal to \(x\). This means if you have a number that isn't a perfect integer, you must find the integer directly below it.

So, for the problem where \(f(x) = [x]\) and we need to find \(f\left(-1\frac{3}{4}\right)\), use the definition of the function:

• First, realize \(-1\frac{3}{4}\) is the same as \(-1.75\).
• Identify integers around this decimal: \(-2\) and \(-1\).
• Select the greatest integer less than or equal to \(-1.75\), which is \(-2\).

Thus, the step by step logic guides you to evaluate that \(f\left(-1\frac{3}{4}\right) = -2\).

Evaluating Functions

When dealing with functions, specifically evaluating them, the goal is to substitute the given input into the function and simplify as needed to find the output. Evaluating functions can range from simple arithmetic to more complex manipulations depending on the type of function.

In this case, we focus on a particular kind of function, the greatest integer function. The task requires identifying and substituting specific input values into the given function definition.

For the greatest integer function, tasks such as determining \(f\left(-1\frac{3}{4}\right)\) involve:

• Recognizing \(x\) within the expression \(-1\frac{3}{4}\).
• Applying the function rule \(f(x) = [x]\).
• Simplifying to find the greatest integer that is less than or equal to \(x\).

Following these steps simplifies the process and helps efficiently evaluate the function.

Integer Functions

Integer functions, like the greatest integer function \(\[x\]\), are a special class of functions that work directly with integers. These functions involve operations that map rational numbers to integers, often rounding down to the nearest integer.

Understanding integer functions involves:

• Recognizing their notation and how it affects numbers: \([x]\) specifically represents the floor function, which rounds \(x\) down.
• Knowing their practical implications: such as in digital computers where these functions help simplify complex calculations that must yield whole numbers.

The greatest integer function is crucial because it provides a consistent way to handle inputs that aren't perfect integers by defining a clear rule of rounding. In the problem we solved, this meant evaluating to find \(f\left(-1\frac{3}{4}\right) = -2\), a straightforward application, illustrating the function’s foundational role in mathematics.