To determine the sum of the functions [tex]\( f(x) = \sqrt{x - 3} \)[/tex] and [tex]\( g(x) = \sqrt{4x - 12} + 15 \)[/tex], we will add these two functions together and then simplify the result.
First, write down the given functions:
[tex]\[ f(x) = \sqrt{x - 3} \][/tex]
[tex]\[ g(x) = \sqrt{4x - 12} + 15 \][/tex]
1. Sum the functions:
[tex]\[(g + f)(x) = \sqrt{x - 3} + \sqrt{4x - 12} + 15\][/tex]
2. Simplify the expression:
Notice that:
[tex]\[ \sqrt{4x - 12} \][/tex]
can be simplified by factoring out a common term:
[tex]\[ 4x - 12 = 4(x - 3) \][/tex]
Thus:
[tex]\[ \sqrt{4x - 12} = \sqrt{4(x - 3)} = 2\sqrt{x - 3} \][/tex]
3. Rewrite and combine like terms:
Replace [tex]\( \sqrt{4x - 12} \)[/tex] with [tex]\( 2\sqrt{x - 3} \)[/tex] in the sum:
[tex]\[ (g + f)(x) = \sqrt{x - 3} + 2\sqrt{x - 3} + 15 \][/tex]
Combine the terms involving [tex]\(\sqrt{x - 3}\)[/tex]:
[tex]\[ (g + f)(x) = 3\sqrt{x - 3} + 15 \][/tex]
So, the sum of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] is:
[tex]\[ (g + f)(x) = 3\sqrt{x - 3} + 15 \][/tex]
Thus, the correct answer is:
[tex]\[ (g+f)(x) = 3\left[(x-3)^{\frac{1}{2}}+5\right] \][/tex]
Therefore, the correct choices for the blank spaces in the expression [tex]\((g+f)(x) = 3\left[(x-\quad \nabla) \frac{1}{2}+\quad \nabla\right]\)[/tex] are:
- For the first blank [tex]\(\nabla\)[/tex]: 3
- For the second blank [tex]\(\nabla\)[/tex]: 5
