Suppose that the function [tex]f[/tex] is defined on the interval [tex]\((-2.5, 1.5)\)[/tex] as follows:
[tex]\[
f(x)=\left\{\begin{array}{ll}
-2 & \text{if } -2.5 \ \textless \ x \leq -1.5 \\
-1 & \text{if } -1.5 \ \textless \ x \leq -0.5 \\
0 & \text{if } -0.5 \ \textless \ x \ \textless \ 0.5 \\
1 & \text{if } 0.5 \leq x \ \textless \ 1.5
\end{array}\right.
\][/tex]

Find [tex]\(f(-1.5), f(-1.4),\)[/tex] and [tex]\(f(0.5)\)[/tex].

[tex]\[
f(-1.5) = \, \square
\][/tex]
[tex]\[
f(-1.4) = \, \square
\][/tex]
[tex]\[
f(0.5) = \, \square
\][/tex]



Answer :

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]