To find the value of [tex]\( f(19) \)[/tex] for the function [tex]\( f(x) = \frac{3}{x+2} - \sqrt{x - 3} \)[/tex], we start by substituting [tex]\( x = 19 \)[/tex] into the given function:
[tex]\[ f(x) = \frac{3}{x+2} - \sqrt{x-3} \][/tex]
Substituting [tex]\( x = 19 \)[/tex]:
[tex]\[ f(19) = \frac{3}{19+2} - \sqrt{19-3} \][/tex]
First, simplify the terms inside the function:
1. Calculate [tex]\( 19 + 2 \)[/tex]:
[tex]\[ 19 + 2 = 21 \][/tex]
2. Calculate [tex]\( 19 - 3 \)[/tex]:
[tex]\[ 19 - 3 = 16 \][/tex]
Next, substitute these simplified values back into the function:
[tex]\[ f(19) = \frac{3}{21} - \sqrt{16} \][/tex]
Now, compute each part separately:
1. Calculate [tex]\( \frac{3}{21} \)[/tex]:
[tex]\[ \frac{3}{21} = \frac{1}{7} \][/tex]
2. Calculate [tex]\( \sqrt{16} \)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
Therefore, the function becomes:
[tex]\[ f(19) = \frac{1}{7} - 4 \][/tex]
Converting [tex]\( \frac{1}{7} \)[/tex] into a decimal for simplicity:
[tex]\[ \frac{1}{7} \approx 0.142857 \][/tex]
So, substituting back:
[tex]\[ f(19) = 0.142857 - 4 \][/tex]
Finally, subtract 4 from the decimal value:
[tex]\[ 0.142857 - 4 = -3.857142857142857 \][/tex]
Thus, the complete statement is:
[tex]\[ f(19) = -3.857142857142857 \][/tex]
