To determine which input value produces the same output for the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we need to compare their values at specific points. The problem provides the function definitions [tex]\( f(x) = -5x + 2 \)[/tex] and [tex]\( g(x) = 2x - 3 \)[/tex], and asks us to check the input values [tex]\(-2, -1, 1, \)[/tex] and [tex]\(2\)[/tex].
Let's evaluate [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at each of these points and see if any results match:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = -5(-2) + 2 = 10 + 2 = 12
\][/tex]
[tex]\[
g(-2) = 2(-2) - 3 = -4 - 3 = -7
\][/tex]
The values don't match, so [tex]\(-2\)[/tex] is not a solution.
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = -5(-1) + 2 = 5 + 2 = 7
\][/tex]
[tex]\[
g(-1) = 2(-1) - 3 = -2 - 3 = -5
\][/tex]
The values don't match, so [tex]\(-1\)[/tex] is not a solution.
3. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = -5(1) + 2 = -5 + 2 = -3
\][/tex]
[tex]\[
g(1) = 2(1) - 3 = 2 - 3 = -1
\][/tex]
The values don't match, so 1 is not a solution.
4. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = -5(2) + 2 = -10 + 2 = -8
\][/tex]
[tex]\[
g(2) = 2(2) - 3 = 4 - 3 = 1
\][/tex]
The values don't match, so 2 is not a solution.
After checking all provided values, we see that none of the input values [tex]\(-2, -1, 1,\)[/tex] or [tex]\(2\)[/tex] produce the same output for both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Therefore, the correct conclusion is that there is no such input value among those given that makes both functions produce the same output.
