Question

If \(f(x)=\operatorname{int}\left(\frac{x}{2}\right),\) find (a) \(f(1.2)\) (b) \(f(1.6)\) (c) \(f(-1.8)\)

Short answer

a) 0, b) 0, c) -1

Step-by-step solution

Understanding the Function

The function is given as \(f(x)=\operatorname{int}\left(\frac{x}{2}\right)\). Here, \(\operatorname{int}(y)\) represents the greatest integer less than or equal to \(y\).

Compute \(\frac{1.2}{2}\)

Divide 1.2 by 2 to get \(\frac{1.2}{2} = 0.6\).

Apply the Greatest Integer Function for \(f(1.2)\)

The greatest integer less than or equal to 0.6 is 0. Therefore, \(f(1.2) = 0\).

Compute \(\frac{1.6}{2}\)

Divide 1.6 by 2 to get \(\frac{1.6}{2} = 0.8\).

Apply the Greatest Integer Function for \(f(1.6)\)

The greatest integer less than or equal to 0.8 is 0. Therefore, \(f(1.6) = 0\).

Compute \(\frac{-1.8}{2}\)

Divide -1.8 by 2 to get \(\frac{-1.8}{2} = -0.9\).

Apply the Greatest Integer Function for \(f(-1.8)\)

The greatest integer less than or equal to -0.9 is -1. Therefore, \(f(-1.8) = -1\).

Key concepts

function evaluation

Function evaluation in mathematics refers to the process of determining the value of a function for a particular input. In the given exercise, we have a function defined as follows: \(f(x) = \text{{int}} \left( \frac{x}{2} \right)\), where \(\text{{int}}(y)\) represents the greatest integer function. To evaluate a function, you substitute the given input (x) with a numerical value and then follow the steps of the function formula. For example, if we evaluate the function at \(x = 1.2\), we first substitute \(1.2\) into the function formula, giving us \(f(1.2) = \text{{int}} \left( \frac{1.2}{2} \right)\). This simplifies to \( \text{{int}}(0.6)\). Because 0.6 lies between 0 and 1, the greatest integer function outputs 0, so \(f(1.2) = 0\). This process applies to any input value, whether it be \(f(1.6)\) or \(f(-1.8)\).

division

Division is a basic arithmetic operation where you split a number into equal parts. In the given exercise, division is an integral part of evaluating the function. To understand how division works in this context, let's consider \(f(1.6)\). Starting with the input value 1.6, the first step is to divide this by 2. Mathematically, we write this as \( \frac{1.6}{2} \), which simplifies to 0.8. Here, we broke down 1.6 into two equal parts, each part being 0.8. Similarly, for \(f(-1.8)\), the input value of -1.8 is divided by 2, yielding -0.9. Division is also important because it often helps to simplify complex functions and equations. Practicing division helps to understand and solve various mathematical problems more easily.

integer part function

The integer part function, also known as the greatest integer function, is a mathematical function that maps a real number to the largest integer less than or equal to it. Denoted as \(\text{{int}}(y)\), it 'rounds down' the input number to the nearest whole number. For example, \( \text{{int}}(0.6)\) is 0, and \( \text{{int}}(0.8)\) is also 0. This characteristic is essential when applying the function in the given exercise. Let's take \(f(-1.8)\), where we end up with the operation \( \text{{int}}(-0.9)\). Though -0.9 is closer to 0 than to -1, the greatest integer that is less than or equal to -0.9 is -1. Thus, \( \text{{int}}(-0.9) = -1\). Understanding the integer part function is crucial because it is commonly used in a wide range of mathematical and real-world applications to manage and interpret continuous data.