To find which expression is equivalent to [tex]\((g - f)(3)\)[/tex] where [tex]\(f(x) = 4 - x^2\)[/tex] and [tex]\(g(x) = 6x\)[/tex], let's go through the following steps:
1. Calculate [tex]\(g(3)\)[/tex]:
[tex]\[
g(3) = 6 \cdot 3 = 18
\][/tex]
2. Calculate [tex]\(f(3)\)[/tex]:
[tex]\[
f(3) = 4 - 3^2 = 4 - 9 = -5
\][/tex]
3. Find [tex]\((g - f)(3)\)[/tex]:
[tex]\[
(g - f)(3) = g(3) - f(3)
\][/tex]
Substituting the values we found:
[tex]\[
(g - f)(3) = 18 - (-5) = 18 + 5 = 23
\][/tex]
Now we need to compare the given expressions to find out which one matches 23 when evaluated.
- First option: [tex]\(6-3-(4+3)^2\)[/tex]
[tex]\[
6 - 3 - (4 + 3)^2 = 6 - 3 - 7^2 = 6 - 3 - 49 = 3 - 49 = -46
\][/tex]
This does not match 23.
- Second option: [tex]\(6-3-\left(4-3^2\right)\)[/tex]
[tex]\[
6 - 3 - (4 - 3^2) = 6 - 3 - (4 - 9) = 6 - 3 - (-5) = 6 - 3 + 5 = 3 + 5 = 8
\][/tex]
This does not match 23.
- Third option: [tex]\(6(3)-4+3^2\)[/tex]
[tex]\[
6 \cdot 3 - 4 + 3^2 = 18 - 4 + 9 = 14 + 9 = 23
\][/tex]
This matches [tex]\(23\)[/tex].
- Fourth option: [tex]\(6(3)-4-3^2\)[/tex]
[tex]\[
6 \cdot 3 - 4 - 3^2 = 18 - 4 - 9 = 18 - 13 = 5
\][/tex]
This does not match 23.
Therefore, the expression [tex]\(6(3)-4+3^2\)[/tex] correctly matches [tex]\((g - f)(3) = 23\)[/tex].
