Let's carefully evaluate the function \( y = \lfloor x \rfloor - 1 \) at the given points and determine which pair of function values are equivalent.
1. Evaluate \( f(-1) \) and \( f(-2) \):
- For \( f(-1) \):
[tex]\[
f(-1) = \lfloor -1 \rfloor - 1 = -1 - 1 = -2
\][/tex]
- For \( f(-2) \):
[tex]\[
f(-2) = \lfloor -2 \rfloor - 1 = -2 - 1 = -3
\][/tex]
Comparing \( f(-1) \) and \( f(-2) \):
[tex]\[
-2 \neq -3 \quad \Rightarrow \quad f(-1) \text{ is not equivalent to } f(-2)
\][/tex]
2. Evaluate \( f(1) \) and \( f(0) \):
- For \( f(1) \):
[tex]\[
f(1) = \lfloor 1 \rfloor - 1 = 1 - 1 = 0
\][/tex]
- For \( f(0) \):
[tex]\[
f(0) = \lfloor 0 \rfloor - 1 = 0 - 1 = -1
\][/tex]
Comparing \( f(1) \) and \( f(0) \):
[tex]\[
0 \neq -1 \quad \Rightarrow \quad f(1) \text{ is not equivalent to } f(0)
\][/tex]
3. Evaluate \( f(-3) \) and \( f(-2.1) \):
- For \( f(-3) \):
[tex]\[
f(-3) = \lfloor -3 \rfloor - 1 = -3 - 1 = -4
\][/tex]
- For \( f(-2.1) \):
[tex]\[
f(-2.1) = \lfloor -2.1 \rfloor - 1 = -3 - 1 = -4
\][/tex]
Comparing \( f(-3) \) and \( f(-2.1) \):
[tex]\[
-4 = -4 \quad \Rightarrow \quad f(-3) \text{ is equivalent to } f(-2.1)
\][/tex]
4. Evaluate \( f(2) \) and \( f(1.9) \):
- For \( f(2) \):
[tex]\[
f(2) = \lfloor 2 \rfloor - 1 = 2 - 1 = 1
\][/tex]
- For \( f(1.9) \):
[tex]\[
f(1.9) = \lfloor 1.9 \rfloor - 1 = 1 - 1 = 0
\][/tex]
Comparing \( f(2) \) and \( f(1.9) \):
[tex]\[
1 \neq 0 \quad \Rightarrow \quad f(2) \text{ is not equivalent to } f(1.9)
\][/tex]
In summary, out of the pairs evaluated, the only equivalent pair is \( f(-3) \) and \( f(-2.1) \):
[tex]\[
f(-3) \text{ is equivalent to } f(-2.1)
\][/tex]
