Problem 6 - Algebra and Trigonometry Real Mathematics, Real People

Question

Given \(f(x)=[x],\) in what interval does \(f(x)=5 ?\)

Step-by-Step Solution

Answer

The interval on which the function \( f(x) = [x] \) is equal to 5 is \( x \) belongs to \( [5, 6) \).

Step 1: Understand the function

When the function \( f(x) = [x] \) is provided, it means that the function outputs the greatest integer less or equal to the input \( x \). For example, \( [2.3] = 2 \), \( [-2.6] = -3 \). The output is always an integer.

Step 2: Identify the interval for the desired output

In this problem, the function value we are interested in is 5. We know from how the greatest integer function works, the function value remains the same (5 in this case) for all \( x \) values between the integer and the next integer. This gives us the interval on the x-axis.

Step 3: Write the solution

Therefore, the interval required is \( x \) belongs to \( [5, 6) \) (inclusive of 5 and exclusive of 6) as all \( x \) in this interval when put into the function will yield an output of 5.

Key Concepts

Step Function

The greatest integer function is a classic example of a step function. A step function is a type of mathematical function that increases or decreases in sudden jumps or steps rather than smoothly over a range of inputs. This function is defined so that it remains constant within specific intervals. In the context of the greatest integer function, the output is the largest integer less than or equal to the input value.

For instance, if you input a number like 2.7 into the greatest integer function, the output will be 2. This happens because 2 is the greatest integer that is less than or equal to 2.7. Similarly, for any input between 5 and 6 (such as 5.2 or 5.9), the output will stay at 5, as no extra steps are required until the threshold of the next integer, which is 6, is reached. Notice that with step functions like this, the graph will resemble a staircase, with horizontal line segments representing the constant output within the intervals.

Piecewise Function

A piecewise function is a function defined by multiple sub-functions, each applicable over certain intervals of the domain. The greatest integer function can be considered a piecewise function because it behaves differently based on the range of input values. Each input yielding a different integer mathematically implies a different formula applied piecewise.

In more in-depth terms, each segment of the greatest integer function is defined by the interval of input values producing a particular integer output. For example, in the range \([5, 6)\), the greatest integer function consistently returns 5, making it appear as though a unique piece or part of the function rule is applied exclusively over this particular range.

These piecewise definitions ensure clarity in understanding how inputs are transformed to outputs based on their intervals.

Interval Notation

Interval notation is a concise method of representing subsets of the real number line. It uses brackets to describe numbers between a start and end point, making it efficient to communicate the domains or outputs of mathematical functions. In interval notation, the brackets are key indicators of whether an endpoint is included or excluded from the interval.

When addressing the greatest integer function in the context of the problem, the interval notation [5, 6) describes all real numbers between 5 and 6, including 5 itself but not 6. This is because, in the greatest integer function, any real number from 5 up to but not including 6 will return 5 upon evaluation.

The square bracket, \([5,\) indicates that 5 is included in the interval, while the parenthesis \(),6)\) signifies exclusion of 6. This notation helps succinctly express the concept that within these bounds, the output remains constant.