To solve the problem, we need to evaluate the function [tex]\(f(x) = \sqrt{x^2 + 3}\)[/tex] at two specific points, [tex]\(x = 2\)[/tex] and [tex]\(x = 5\)[/tex], and then find the sum of these evaluations.
### Step 1: Calculate [tex]\( f(2) \)[/tex]
First, substitute [tex]\(x = 2\)[/tex] into the function:
[tex]\[ f(2) = \sqrt{2^2 + 3} \][/tex]
Simplify the expression inside the square root:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 4 + 3 = 7 \][/tex]
So,
[tex]\[ f(2) = \sqrt{7} \][/tex]
The approximate value of [tex]\(\sqrt{7}\)[/tex] is:
[tex]\[ f(2) \approx 2.6457513110645907 \][/tex]
### Step 2: Calculate [tex]\( f(5) \)[/tex]
Next, substitute [tex]\(x = 5\)[/tex] into the function:
[tex]\[ f(5) = \sqrt{5^2 + 3} \][/tex]
Simplify the expression inside the square root:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 25 + 3 = 28 \][/tex]
So,
[tex]\[ f(5) = \sqrt{28} \][/tex]
The approximate value of [tex]\(\sqrt{28}\)[/tex] is:
[tex]\[ f(5) \approx 5.291502622129181 \][/tex]
### Step 3: Find [tex]\( f(2) + f(5) \)[/tex]
Finally, add the two evaluated values:
[tex]\[ f(2) + f(5) = 2.6457513110645907 + 5.291502622129181 \][/tex]
[tex]\[ f(2) + f(5) \approx 7.937253933193772 \][/tex]
Thus, the sum [tex]\(f(2) + f(5) \approx 7.937253933193772\)[/tex].
