To find the input value that produces the same output for the functions [tex]\( f(x) = -0.5x + 2 \)[/tex] and [tex]\( g(x) = 2x - 3 \)[/tex], you need to compare the outputs of both functions for the same input values.
We are given:
[tex]\[ f(x) = -0.5x + 2 \][/tex]
[tex]\[ g(x) = 2x - 3 \][/tex]
From the table for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-3 & 3.5 \\
-2 & 3 \\
-1 & 2.5 \\
0 & 2 \\
1 & 1.5 \\
2 & 1 \\
3 & 0.5 \\
\hline
\end{array}
\][/tex]
Next, let's calculate [tex]\( g(x) \)[/tex] for the same values of [tex]\( x \)[/tex]:
For [tex]\( x = -3 \)[/tex]:
[tex]\[ g(-3) = 2(-3) - 3 = -6 - 3 = -9 \][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 2(-2) - 3 = -4 - 3 = -7 \][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = 2(-1) - 3 = -2 - 3 = -5 \][/tex]
For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 2(0) - 3 = 0 - 3 = -3 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 2(1) - 3 = 2 - 3 = -1 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 2(2) - 3 = 4 - 3 = 1 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = 2(3) - 3 = 6 - 3 = 3 \][/tex]
Now we can compare the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] for the same input values:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
-3 & 3.5 & -9 \\
-2 & 3 & -7 \\
-1 & 2.5 & -5 \\
0 & 2 & -3 \\
1 & 1.5 & -1 \\
2 & 1 & 1 \\
3 & 0.5 & 3 \\
\hline
\end{array}
\][/tex]
Looking at the table, we see that [tex]\( f(x) = g(x) \)[/tex] when [tex]\( x = 2 \)[/tex]:
For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 1 \][/tex]
[tex]\[ g(2) = 1 \][/tex]
Therefore, the input value that produces the same output value in both charts is [tex]\( 2 \)[/tex].
So the correct answer is:
[tex]\[ \boxed{2} \][/tex]
