Problem 51 - Precalculus with Limits

Question

In Exercises 51-56, sketch the graph of the function. \(g(x) = - [[x]]\)

Step-by-Step Solution

Answer

The graph is a stair-shaped curve, with a negative slope. It jumps down at every integer value of \(x\).

Step 1: Understanding the Floor Function

Identify the output of the floor function by giving it some values. The function \([[x]]\) will output the largest whole number less than or equal to \(x\). This means, for any given integer \(x\), the function will output \(x\). However, for any non-integer value \(x\), it will round down to the nearest integer. For example, \([[1.2]] = 1\), \([[5]] = 5\), and \([[3.8]] = 3\).

Step 2: Applying the Negation

Since the function is \(g(x) = - [[x]]\), the outputs from \([[x]]\) will be negated. This means the output value will be the negative of the value from step 1. For example, if \(x = 1.2\), then \(g(x) = - [[1.2]] = -1\). If \(x = 5\), then \(g(x) = - [[5]] = -5\). And if \(x = 3.8\), then \(g(x) = - [[3.8]] = -3\).

Step 3: Sketching the Graph

Start plotting some points on a graph. Recognize the pattern of step lines between each integer \(x\)-value. At each integer, the function jumps down to the next lower value. The graph is a series of steps that decrease in value at every integer value of \(x\). The plotted points create a negatively sloped stair-shaped curve.

Key Concepts

floor function

The floor function is a special function that rounds a number down to the nearest whole number. This means that regardless of the decimal part of a given number, it takes the largest integer that is less than or equal to the original number.
For example:

• For 3.7, the floor function \([3.7]\) equals 3, since 3 is the largest integer less than or equal to 3.7.
• For -2.3, the floor function \([-2.3]\) equals -3, as the nearest lower integer is -3.

Understanding this behavior helps when graphing functions that involve rounding, as each non-integer value maps to the same integer outcome.

step graph

A step graph is a visual representation of a function like the floor function. It appears as a series of horizontal line segments, giving it a characteristic stair-step appearance.
Here's what you need to know:

• Each step represents a range of input values that all produce the same output value. This is due to the "jump" at each integer value.
• In the case of the graph for the function \(g(x) = - [[x]]\), each step begins just after an integer and continues until the next integer, where a sudden drop or step to the next line segment occurs.

When sketching a step graph, you'll create horizontal lines that drop at each whole number, resulting in a staircase shape that descends along the x-axis.

integer function

An integer function simplifies numbers by converting them into integers. It's like reducing a complex form into something simple, using integers.
Consider these points:

• The floor function is a type of integer function because it outputs integers by taking the nearest lower integer from any real number.
• Doing this repeatedly across numbers forms a sequence of integers, each "step" mapping a range of real numbers to a specific integer output.

Integer functions often display non-continuous graphs. This non-continuity is visible in the breaks or jumps at each integer point, making them useful in real-life scenarios where measurements must be whole numbers.