To determine which expression is equivalent to [tex]\((g - f)(3)\)[/tex], let's first break down and evaluate the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at [tex]\( x = 3 \)[/tex].
Given the functions:
[tex]\[ f(x) = 4 - x^2 \][/tex]
[tex]\[ g(x) = 6x \][/tex]
First, we calculate [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = 4 - (3)^2 \][/tex]
[tex]\[ f(3) = 4 - 9 \][/tex]
[tex]\[ f(3) = -5 \][/tex]
Next, we calculate [tex]\( g(3) \)[/tex]:
[tex]\[ g(3) = 6 \cdot 3 \][/tex]
[tex]\[ g(3) = 18 \][/tex]
Now, we need to find [tex]\((g - f)(3)\)[/tex], which is:
[tex]\[ (g - f)(3) = g(3) - f(3) \][/tex]
[tex]\[ (g - f)(3) = 18 - (-5) \][/tex]
[tex]\[ (g - f)(3) = 18 + 5 \][/tex]
[tex]\[ (g - f)(3) = 23 \][/tex]
We observe the expressions that are provided:
1. [tex]\(6 - 3 - (4 + 3)^2\)[/tex]
2. [tex]\(6 - 3 - \left(4 - 3^2\right)\)[/tex]
3. [tex]\(6(3) - 4 + 3^2\)[/tex]
4. [tex]\(6(3) - 4 - 3^2\)[/tex]
To find the correct expression, let's evaluate each one:
1. [tex]\(6 - 3 - (4 + 3)^2 = 6 - 3 - 7^2 = 6 - 3 - 49 = 3 - 49 = -46\)[/tex]
2. [tex]\(6 - 3 - \left(4 - 3^2\right) = 6 - 3 - (4 - 9) = 6 - 3 - (-5) = 6 - 3 + 5 = 8\)[/tex]
3. [tex]\(6(3) - 4 + 3^2 = 18 - 4 + 9 = 18 - 4 + 9 = 14 + 9 = 23\)[/tex]
4. [tex]\(6(3) - 4 - 3^2 = 18 - 4 - 9 = 18 - 4 - 9 = 14 - 9 = 5\)[/tex]
The correct expression equivalent to [tex]\((g - f)(3)\)[/tex] is:
[tex]\[ 6(3) - 4 + 3^2 \][/tex]
Thus, the correct answer is:
[tex]\[ 6(3) - 4 + 3^2 \][/tex]
