To determine which input value produces the same output value for the given two functions, we need to evaluate both functions at the given input values and compare the results.
Here are the two functions:
1. [tex]\( f_1(x) = 2x + 3 \)[/tex]
2. [tex]\( f_2(x) = x^2 - x - 6 \)[/tex]
We will check each input value [tex]\((-12, -9, -6, -3)\)[/tex] to see if [tex]\( f_1(x) = f_2(x) \)[/tex].
### For [tex]\( x = -12 \)[/tex]:
[tex]\[ f_1(-12) = 2(-12) + 3 = -24 + 3 = -21 \][/tex]
[tex]\[ f_2(-12) = (-12)^2 - (-12) - 6 = 144 + 12 - 6 = 150 \][/tex]
Since [tex]\( f_1(-12) \neq f_2(-12) \)[/tex], -12 is not a valid input.
### For [tex]\( x = -9 \)[/tex]:
[tex]\[ f_1(-9) = 2(-9) + 3 = -18 + 3 = -15 \][/tex]
[tex]\[ f_2(-9) = (-9)^2 - (-9) - 6 = 81 + 9 - 6 = 84 \][/tex]
Since [tex]\( f_1(-9) \neq f_2(-9) \)[/tex], -9 is not a valid input.
### For [tex]\( x = -6 \)[/tex]:
[tex]\[ f_1(-6) = 2(-6) + 3 = -12 + 3 = -9 \][/tex]
[tex]\[ f_2(-6) = (-6)^2 - (-6) - 6 = 36 + 6 - 6 = 36 \][/tex]
Since [tex]\( f_1(-6) \neq f_2(-6) \)[/tex], -6 is not a valid input.
### For [tex]\( x = -3 \)[/tex]:
[tex]\[ f_1(-3) = 2(-3) + 3 = -6 + 3 = -3 \][/tex]
[tex]\[ f_2(-3) = (-3)^2 - (-3) - 6 = 9 + 3 - 6 = 6 \][/tex]
Since [tex]\( f_1(-3) \neq f_2(-3) \)[/tex], -3 is not a valid input.
After checking all the given input values, we see that none of the input values [tex]\((-12, -9, -6, -3)\)[/tex] produce the same output for both functions. Thus, there is no input value from the given set that satisfies [tex]\( f_1(x) = f_2(x) \)[/tex].
