Let's find the values of the function [tex]\( f \)[/tex] at the specified points: [tex]\( f(-1.5) \)[/tex], [tex]\( f(-1.4) \)[/tex], and [tex]\( f(0.5) \)[/tex]. The piecewise function [tex]\( f(x) \)[/tex] is defined as follows:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
-2 & \text { if } -2.5 < x \leq -1.5 \\
-1 & \text { if } -1.5 < x \leq -0.5 \\
0 & \text { if } -0.5 < x < 0.5 \\
1 & \text { if } 0.5 \leq x < 1.5
\end{array}
\right.
\][/tex]
1. Finding [tex]\( f(-1.5) \)[/tex]:
According to the definition of the function:
- For [tex]\( -2.5 < x \leq -1.5 \)[/tex], [tex]\( f(x) = -2 \)[/tex]
- For [tex]\( -1.5 < x \leq -0.5 \)[/tex], [tex]\( f(x) = -1 \)[/tex]
Since [tex]\( -1.5 \)[/tex] is at the boundary and according to the given interval notation, it falls into the first case because it includes [tex]\( -1.5 \)[/tex] in the inequality [tex]\( -2.5 < x \leq -1.5 \)[/tex]. Therefore:
[tex]\[
f(-1.5) = -2
\][/tex]
2. Finding [tex]\( f(-1.4) \)[/tex]:
According to the definition of the function:
- For [tex]\( -2.5 < x \leq -1.5 \)[/tex], [tex]\( f(x) = -2 \)[/tex]
- For [tex]\( -1.5 < x \leq -0.5 \)[/tex], [tex]\( f(x) = -1 \)[/tex]
Since [tex]\( -1.4 \)[/tex] falls within [tex]\( -1.5 < x \leq -0.5 \)[/tex], we have:
[tex]\[
f(-1.4) = -1
\][/tex]
3. Finding [tex]\( f(0.5) \)[/tex]:
According to the definition of the function:
- For [tex]\( -0.5 < x < 0.5 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- For [tex]\( 0.5 \leq x < 1.5 \)[/tex], [tex]\( f(x) = 1 \)[/tex]
Since [tex]\( 0.5 \)[/tex] falls within [tex]\( 0.5 \leq x < 1.5 \)[/tex], we have:
[tex]\[
f(0.5) = 1
\][/tex]
Therefore, the values of the function at the specified points are:
[tex]\[
f(-1.5) = -2
\][/tex]
[tex]\[
f(-1.4) = -1
\][/tex]
[tex]\[
f(0.5) = 1
\][/tex]
