To determine which function maps [tex]\(x = 2\)[/tex] to 32, we'll substitute [tex]\(x = 2\)[/tex] into each function and evaluate the result.
1. Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
[tex]\[
f(x) = -3x^2 - 4
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = -3(2)^2 - 4 = -3(4) - 4 = -12 - 4 = -16
\][/tex]
This does not map [tex]\(x = 2\)[/tex] to 32.
2. Evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
[tex]\[
g(x) = 4(x + 3)^2 - 68
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
g(2) = 4(2 + 3)^2 - 68 = 4(5)^2 - 68 = 4(25) - 68 = 100 - 68 = 32
\][/tex]
This maps [tex]\(x = 2\)[/tex] to 32.
3. Evaluate [tex]\(h(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
[tex]\[
h(x) = 3x
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
h(2) = 3(2) = 6
\][/tex]
This does not map [tex]\(x = 2\)[/tex] to 32.
4. Evaluate [tex]\(j(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
[tex]\[
j(x) = 2x - 62
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
j(2) = 2(2) - 62 = 4 - 62 = -58
\][/tex]
This does not map [tex]\(x = 2\)[/tex] to 32.
After evaluating each function, we see that only [tex]\(g(x)\)[/tex] maps [tex]\(x = 2\)[/tex] to 32. Thus, the function that maps [tex]\(x = 2\)[/tex] to 32 is:
[tex]\[
g(x) = 4(x + 3)^2 - 68
\][/tex]
