To solve the problem of finding the input value [tex]\( x \)[/tex] that produces the same output value for both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we can follow these steps:
Step 1: Determine the expressions for the two functions.
- [tex]\( f(x) = -0.5x + 2 \)[/tex]
- [tex]\( g(x) = 2x - 3 \)[/tex]
Step 2: Evaluate which input [tex]\( x \)[/tex] will make the outputs of both functions equal by exploring the given 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]
Step 3: Now, calculate the corresponding [tex]\( g(x) \)[/tex] values for each [tex]\( x \)[/tex] in the same range:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-3 & 2(-3) - 3 = -6 - 3 = -9 \\
-2 & 2(-2) - 3 = -4 - 3 = -7 \\
-1 & 2(-1) - 3 = -2 - 3 = -5 \\
0 & 2(0) - 3 = 0 - 3 = -3 \\
1 & 2(1) - 3 = 2 - 3 = -1 \\
2 & 2(2) - 3 = 4 - 3 = 1 \\
3 & 2(3) - 3 = 6 - 3 = 3 \\
\hline
\end{array}
\][/tex]
Step 4: Compare outputs [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] for the same [tex]\( x \)[/tex]:
- [tex]\( x = -3 \)[/tex], [tex]\( f(-3) = 3.5 \)[/tex] and [tex]\( g(-3) = -9 \)[/tex] (Not equal)
- [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = 3 \)[/tex] and [tex]\( g(-2) = -7 \)[/tex] (Not equal)
- [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 2.5 \)[/tex] and [tex]\( g(-1) = -5 \)[/tex] (Not equal)
- [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 2 \)[/tex] and [tex]\( g(0) = -3 \)[/tex] (Not equal)
- [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 1.5 \)[/tex] and [tex]\( g(1) = -1 \)[/tex] (Not equal)
- [tex]\( x = 2 \)[/tex], [tex]\( f(2) = 1 \)[/tex] and [tex]\( g(2) = 1 \)[/tex] (Equal)
- [tex]\( x = 3 \)[/tex], [tex]\( f(3) = 0.5 \)[/tex] and [tex]\( g(3) = 3 \)[/tex] (Not equal)
Conclusion: The input value [tex]\( x = 2 \)[/tex] produces the same output value for both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Therefore, the answer is:
2
