To determine which expression is equivalent to [tex]\((g-f)(3)\)[/tex] where [tex]\(f(x) = 4 - x^2\)[/tex] and [tex]\(g(x) = 6x\)[/tex], we need to first evaluate [tex]\(g(3) - f(3)\)[/tex].
1. Calculate [tex]\(g(3)\)[/tex]:
[tex]\[ g(x) = 6x \][/tex]
[tex]\[ g(3) = 6 \times 3 = 18 \][/tex]
2. Calculate [tex]\(f(3)\)[/tex]:
[tex]\[ f(x) = 4 - x^2 \][/tex]
[tex]\[ f(3) = 4 - (3^2) = 4 - 9 = -5 \][/tex]
3. Compute [tex]\((g-f)(3)\)[/tex]:
[tex]\[ (g-f)(3) = g(3) - f(3) \][/tex]
[tex]\[ (g-f)(3) = 18 - (-5) = 18 + 5 = 23 \][/tex]
Now we compare this result to the given options:
1. [tex]\(6 - 3 - (4 + 3)^2\)[/tex]
[tex]\[ 6 - 3 - (4 + 3)^2 = 6 - 3 - 7^2 = 6 - 3 - 49 = -46 \][/tex]
This is not equal to 23.
2. [tex]\(6 - 3 - \left(4 - 3^2\right)\)[/tex]
[tex]\[ 6 - 3 - \left(4 - 3^2\right) = 6 - 3 - (4 - 9) = 6 - 3 - (-5) = 6 - 3 + 5 = 8 \][/tex]
This is not equal to 23.
3. [tex]\(6(3) - 4 + 3^2\)[/tex]
[tex]\[ 6(3) - 4 + 3^2 = 18 - 4 + 9 = 23 \][/tex]
This matches our calculated result.
4. [tex]\(6(3) - 4 - 3^2\)[/tex]
[tex]\[ 6(3) - 4 - 3^2 = 18 - 4 - 9 = 5 \][/tex]
This is not equal to 23.
Thus, the correct expression equivalent to [tex]\((g-f)(3)\)[/tex] is:
[tex]\[ 6(3) - 4 + 3^2 \][/tex]
