Decoding the Depths: A Dive into the Composite Floor Function

Step-by-step Guide to Understand Floor Function

Here is a step-by-step guide to understand floor function:

Step 1: Foundational Pillars

• 1.1. Floor Function, \(⌊x⌋\): This gives the greatest integer less than or equal to x. For instance, \(⌊3.7⌋ = 3\), while \( ⌊-3.7⌋ = -4\).
• 1.2. Linear Transformation: A function of the form \(y=mx+c\), where ‘\(m\)’ and ‘\(c\)’ are constants, represents a straight line. In our composite function, the transformation \(g(x-b)\) can be seen as a shifted and/or scaled version of \(x\).

Step 2: Peeling Off The Layers of \(f(x)=a[g(x−b)]+k\)

• 2.1. Innermost Layer, \(g(x−b)\):
  • The term \(x−b\) signifies a horizontal shift. If ‘\(b\)’ is positive, the graph moves to the right by ‘\(b\)’ units; otherwise, to the left.
  • The function ‘\(g\)’ then acts upon this shifted ‘\(x\)’, which can stretch, compress, or flip the graph depending on its nature.
• 2.2. The Floor Encapsulation, \(⌊g(x-b)⌋\): Post the transformation of \(x−b\) by ‘\(g\)’, the floor function rounds the result down to the nearest integer.
• 2.3. Scaling and Vertical Shift:
  • \(a[…]\): ‘\(a\)’ can stretch or compress the output vertically. If \(|a| > 1\), it stretches; if \(0 < |a| < 1\), it compresses. A negative ‘\(a\)’ will flip the graph upside-down.
  • \(… + k\): This term vertically shifts the entire graph. A positive ‘\(k\)’ will move it up, and a negative ‘\(k\)’ will move it down.

Step 3: An Illustrative Example

Consider \(f(x)=2[3(x−1)]+1\).

• 3.1. Unpacking \(g(x−b)=3(x−1)\): Here, ‘\(b\)’ is \(1\), shifting the graph one unit right. The factor \(3\) stretches it vertically.
• 3.2. Applying the Floor: The function then rounds the values down, creating a step-like graph.
• 3.3. The Scaling and Shifting: Finally, multiply the values by \(2\) (stretching the steps) and shift one unit up.

Step 4: Observing the Effects on Graphs

To deeply understand the function’s behavior, graphing is instrumental. Sketching the transformations step-by-step can elucidate how each parameter influences the function.

Final Words

Our journey through the formula \(f(x)=a[g(x−b)]+k\) showcases the intricacies and depth inherent in mathematical functions. The combination of shifts, scalings, and the floor function’s unique properties culminate in a rich landscape of possible function graphs. Embracing each term’s effect in isolation, and then in combination, empowers us with a thorough grasp of this composite function.

Examples:

Example 1:

Given the function \(f(x)=3[2x+4]\), determine the value at \(x=1.25\).

Solution:

Plug \(1.25\) in the function equation and simplify. So, we have:

\(f(1.25)=3[2x+4]=3[2(1.25)+4]\)

\(3[2.5+4]=3[6.5]=3(6)=18\)

So, the answer is \(18\)

Example 2:

Given the function \(f(x)=[x+7]−4\), determine the value at \(x=−4.5\).

Solution:

Plug \(−4.5\) in the function equation and simplify. So, we have:

\(f(−4.5)=[x+7]-4=[(-4.5)+7]-4\)

\([(-4.5)+7]-4=[2.5]-4=(2)-4=-2\)

So, the answer is \(-2\)