91Ó°ÊÓ

Stepwise graphs are fascinating as they show how functions like the greatest integer function behave by jumping between values instead of following a smooth curve. The greatest integer function, denoted by \( \llbracket x \rrbracket \), is a classic example of a stepwise graph. It assigns to each real number the largest integer less than or equal to it. This results in steps appearing on a graph, where the function stays constant between two consecutive integers and then jumps to the next level.

In the context of our exercise, sketching the graph for \( f(x) = \llbracket x+2 \rrbracket \) involves shifting this stepwise behavior. Similarly, changing the starting point, as seen with \( f(x) = \llbracket x \rrbracket + 2 \), moves the entire graph upward, highlighting how the function's steps are displaced to reflect the transformation applied to \( x \).

• The steps occur at integer values.
• Each step is typically one unit high.
• The function jumps at each integer transition.

Graph transformations involve modifying the shape, position, or size of a graph. For piecewise functions like step functions, these transformations shift, stretch, or compress the step pattern. In various parts of this exercise, different transformations are applied. Understanding these transformations helps in predicting and sketching how the graph of a function changes.

Let's break down some key transformations:

• Vertical Shifts: Adding a constant term translates the graph upwards or downwards. In part (b), \( f(x) = \llbracket x \rrbracket + 2 \), moves every step of the graph up by two units.
• Horizontal Shifts: Adding a constant inside the function shifts the graph left or right. For \( f(x) = \llbracket x+2 \rrbracket \), the graph moves left by two units because of the \(+2\) inside the function \( \llbracket x \rrbracket \).
• Vertical Compression/Stretch: Multiplying the function by a constant changes the step height. In part (c), the presence of \( \frac{1}{2} \) compresses each step's height by half.
• Horizontal Compression/Stretch: Applying a multiplication factor within the \( x \) value compresses or stretches the step intervals. In part (d), multiplying \( x \) by \( \frac{1}{2} \) yields a graph that steps every 2 units along the x-axis, effectively stretching the horizontal scale.

The ceiling function, denoted as \( \lceil x \rceil \), is related to the greatest integer function but handles numbers a bit differently. Instead of finding the largest integer less than or equal to \( x \), it gives the smallest integer greater than or equal to \( x \). This change in operation results in stepwise graphs that differ notably in their placement and direction.

In the exercise part (e), the function \( f(x) = -\llbracket-x\rrbracket \) effectively results in the ceiling function \( f(x) = \lceil x \rceil \). Here's what sets the ceiling function apart:

• Steps occur at each integer value, similar to the floor function, but include the right endpoint instead of the left.
• Each step corresponds to the next integer as \( x \) increases.
• The function behaves like a staircase that never dips but always ascends or moves directly across at each step marker.

The ceiling function's behavior creates a visual contrast to the greatest integer or floor function, as its steps appear at different positions, generating a unique graph pattern.