To find \( f(19) \) given the function \( f(x) = \frac{3}{x+2} - \sqrt{x-3} \), we will substitute \( x = 19 \) into the function and simplify.
1. Substitute \( x = 19 \) into the function:
[tex]\[
f(19) = \frac{3}{19 + 2} - \sqrt{19 - 3}
\][/tex]
2. Simplify the denominator in the fraction:
[tex]\[
19 + 2 = 21
\][/tex]
So,
[tex]\[
f(19) = \frac{3}{21} - \sqrt{19 - 3}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{3}{21} = \frac{1}{7}
\][/tex]
So,
[tex]\[
f(19) = \frac{1}{7} - \sqrt{19 - 3}
\][/tex]
4. Simplify the expression inside the square root:
[tex]\[
19 - 3 = 16
\][/tex]
Thus,
[tex]\[
f(19) = \frac{1}{7} - \sqrt{16}
\][/tex]
5. Find the square root of 16:
[tex]\[
\sqrt{16} = 4
\][/tex]
Therefore,
[tex]\[
f(19) = \frac{1}{7} - 4
\][/tex]
6. Convert 4 to a fraction with a denominator of 7 to combine the terms:
[tex]\[
4 = \frac{28}{7}
\][/tex]
Hence,
[tex]\[
f(19) = \frac{1}{7} - \frac{28}{7}
\][/tex]
7. Subtract the fractions:
[tex]\[
f(19) = \frac{1 - 28}{7} = \frac{-27}{7}
\][/tex]
8. Simplify the fraction:
[tex]\[
\frac{-27}{7} = -3.857142857142857
\][/tex]
Thus, the value of \( f(19) \) is:
[tex]\[
f(19) = -3.857142857142857
\][/tex]
