Let us explore the expressions and determine the one that matches the correct result for [tex]\((g - f)(3)\)[/tex] where [tex]\(f(x) = 4 - x^2\)[/tex] and [tex]\(g(x) = 6x\)[/tex].
First, we need to evaluate [tex]\(f(3)\)[/tex] and [tex]\(g(3)\)[/tex] separately:
1. Evaluating [tex]\(f(3)\)[/tex]:
[tex]\[
f(x) = 4 - x^2 \Rightarrow f(3) = 4 - 3^2 = 4 - 9 = -5
\][/tex]
2. Evaluating [tex]\(g(3)\)[/tex]:
[tex]\[
g(x) = 6x \Rightarrow g(3) = 6 \cdot 3 = 18
\][/tex]
Next, we need to find [tex]\((g - f)(3)\)[/tex]:
[tex]\[
(g-f)(3) = g(3) - f(3) = 18 - (-5) = 18 + 5 = 23
\][/tex]
Now, let's check each given expression to see which one correctly evaluates to [tex]\(23\)[/tex]:
- Expression 1: [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 the required value of [tex]\(23\)[/tex].
- Expression 2: [tex]\(6 - 3 - \left(4 - 3^2\right)\)[/tex]
[tex]\[
6 - 3 - \left(4 - 3^2\right) = 6 - 3 - (4 - 9) = 6 - 3 - (-5) = 6 - 3 + 5 = 3 + 5 = 8
\][/tex]
This does not match the required value of [tex]\(23\)[/tex].
- Expression 3: [tex]\(6(3) - 4 + 3^2\)[/tex]
[tex]\[
6(3) - 4 + 3^2 = 18 - 4 + 9 = 18 - 4 + 9 = 14 + 9 = 23
\][/tex]
This matches the required value of [tex]\(23\)[/tex].
- Expression 4: [tex]\(6(3) - 4 - 3^2\)[/tex]
[tex]\[
6(3) - 4 - 3^2 = 18 - 4 - 9 = 18 - 4 - 9 = 14 - 9 = 5
\][/tex]
This does not match the required value of [tex]\(23\)[/tex].
Therefore, the correct expression is:
[tex]\[
6(3) - 4 + 3^2
\][/tex]
So, the equivalent expression to [tex]\((g - f)(3)\)[/tex] is:
[tex]\[
6(3) - 4 + 3^2
\][/tex]
