Problem 77 - College Algebra

Question

In Exercises, if \(f(x)=\) int \((x),\) find each function value. $$f(1.06)$$

Step-by-Step Solution

Answer

The function value \(f(1.06)\) is 1.

Step 1: Understanding the Integer Function

The integer function, denoted as 'int', is a step function. It rounds down a real number to the largest integer less than or equal to the number. As such, for any positive real number x such that n ≤ x < n + 1, where n is an integer, int(x) returns n. Similarly for a negative real number, int(x) returns the value of the smallest integer greater than the number.

Step 2: Apply the Integer Function to Given Value

In the case of \(f(1.06)\), since 1 ≤ 1.06 < 2, the greatest integer less than or equal to 1.06 is 1. Therefore, int(1.06) = 1.

Key Concepts

Step Function

Imagine a staircase, each step representing a different integer. A step function is like the flat surface of each step, where no matter where you stand, the elevation doesn't change until you take a step up or down to a new flat surface. In mathematics, a step function maps every number in an interval to a single value, much like each flat step. This creates a graph that appears as constant horizontal lines, or 'steps', that jump to a new value at specified intervals.

A classic example of a step function is the integer function, which assigns to every real number the greatest integer that is less than or equal to it. When graphed, the step function remains constant until it reaches the next integer, where it 'jumps' up or down depending on the direction of the graph. This makes the step function highly useful in applications where value increments occur at discrete intervals, such as quantizing measurements or modeling scenarios where changes occur in distinct steps rather than gradually.

Rounding Down Real Numbers

In mathematics, rounding down is a way to convert a real number to the largest integer that is not greater than the original number. This is done by 'truncating' the number, or essentially, removing all the digits after the decimal point without rounding them. The result is the integer component of the number, without any fractional part.

The process of rounding down real numbers is essential when discrete values are needed, such as counting items, slotting times on a schedule, or allocating resources. When you round down, you're engaging in a conservative estimation, ensuring that the result does not exceed the original value. This is vitally important in settings where overestimating could lead to negative consequences, like budgeting or planning material requirements.

Greatest Integer Function

The greatest integer function, also known as the 'floor function' or by the notation 'int', is a particular type of step function that takes any real number and gives back the greatest integer less than or equal to that number. For any positive number, this means 'rounding down' to the nearest whole number. In the case of negative numbers, since they are to the left of zero on the number line, rounding down means going to the next more negative whole number.

When dealing with the greatest integer function using a value like 1.06, we identify the greatest integer that doesn't surpass the number - in this case, 1. This function is not only a theoretical construct; it has practical applications in computer science for algorithms that need to partition data into integer-based categories or determine array indices from real number coordinates.

Piecewise Functions

A piecewise function is a type of function composed of multiple sub-functions, each of which applies to a certain interval of the main function's domain. Basically, it's like having different rules for different parts of the number line. Each sub-function is defined over a specific interval and the function 'switches' between these sub-functions depending on the input.

Step functions are an example of piecewise functions, which are defined by different constants over different intervals. This allows for greater flexibility, enabling the modeling of scenarios where the relationship between variables changes abruptly at certain points, such as tax brackets, shipping rates based on weight, or bulk pricing in sales. Piecewise functions can be written using a brace to show that the function takes different forms over different intervals, providing a clear and organized way to display the function's behavior over its entire domain.