# Decoding the Depths: A Dive into the Composite Floor Function

The realm of mathematical functions houses various unique and intriguing functions. The floor function, often denoted by $$⌊ ⌋$$, is a fascinating function that yields the largest integer less than or equal to a given number. Our discussion today encapsulates a composite floor function with parameters. Let's dissect $$f(x)=a[g(x−b)]+k$$ and unfurl its nuances. ## 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 $$-2$$

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