Sure, let's evaluate the floor function [tex]\( f(x) = \lfloor x \rfloor \)[/tex] for the given input values step-by-step.
### Step 1: Understanding the Floor Function
The floor function, [tex]\( \lfloor x \rfloor \)[/tex], takes a real number [tex]\( x \)[/tex] and maps it to the greatest integer less than or equal to [tex]\( x \)[/tex]. Essentially, you round down [tex]\( x \)[/tex] to the nearest integer.
### Step 2: Evaluating [tex]\( f(2) \)[/tex]
We need to find [tex]\( f(2) \)[/tex]. Since 2 is already an integer, the greatest integer less than or equal to 2 is 2 itself.
[tex]\[
f(2) = \lfloor 2 \rfloor = 2
\][/tex]
### Step 3: Evaluating [tex]\( f(6.8) \)[/tex]
Next, we need to find [tex]\( f(6.8) \)[/tex]. The number 6.8 lies between 6 and 7. The greatest integer less than or equal to 6.8 is 6.
[tex]\[
f(6.8) = \lfloor 6.8 \rfloor = 6
\][/tex]
### Step 4: Evaluating [tex]\( f(-3.3) \)[/tex]
Lastly, we need to find [tex]\( f(-3.3) \)[/tex]. The number -3.3 lies between -4 and -3. Since we are looking for the greatest integer less than or equal to -3.3, it is -4.
[tex]\[
f(-3.3) = \lfloor -3.3 \rfloor = -4
\][/tex]
### Summary of Results
[tex]\[
\begin{array}{l}
f(2) = 2 \\
f(6.8) = 6 \\
f(-3.3) = -4
\end{array}
\][/tex]
Therefore, the evaluations of the floor function for the given input values are:
[tex]\[
f(2) = 2, \quad f(6.8) = 6, \quad f(-3.3) = -4
\][/tex]
