To determine which function maps [tex]\( x = 2 \)[/tex] to [tex]\( 32 \)[/tex], we'll evaluate each function at [tex]\( x = 2 \)[/tex] and see which one results in [tex]\( 32 \)[/tex].
1. Evaluating [tex]\( f(x) = -3x^2 - 4 \)[/tex]:
[tex]\[
f(2) = -3(2)^2 - 4 = -3 \cdot 4 - 4 = -12 - 4 = -16
\][/tex]
Hence, [tex]\( f(2) = -16 \)[/tex].
2. Evaluating [tex]\( g(x) = 4(x + 3)^2 - 68 \)[/tex]:
[tex]\[
g(2) = 4(2 + 3)^2 - 68 = 4 \cdot 5^2 - 68 = 4 \cdot 25 - 68 = 100 - 68 = 32
\][/tex]
Hence, [tex]\( g(2) = 32 \)[/tex].
3. Evaluating [tex]\( h(x) = 3x \)[/tex]:
[tex]\[
h(2) = 3 \cdot 2 = 6
\][/tex]
Hence, [tex]\( h(2) = 6 \)[/tex].
4. Evaluating [tex]\( j(x) = 2x - 62 \)[/tex]:
[tex]\[
j(2) = 2 \cdot 2 - 62 = 4 - 62 = -58
\][/tex]
Hence, [tex]\( j(2) = -58 \)[/tex].
Among the evaluated results, only the function [tex]\( g(x) = 4(x + 3)^2 - 68 \)[/tex] maps [tex]\( x = 2 \)[/tex] to [tex]\( 32 \)[/tex]. Therefore, [tex]\( \boxed{g(x)} \)[/tex] is the function where [tex]\( x = 2 \)[/tex] is mapped to [tex]\( 32 \)[/tex].
