Sure! Let's solve for each function with the specified inputs individually:
1. Evaluating [tex]\( f(-2) \)[/tex]:
The function [tex]\( f(x) \)[/tex] is given by:
[tex]\[
f(x) = \sqrt{-6x + 19}
\][/tex]
Plugging [tex]\( x = -2 \)[/tex] into the function, we get:
[tex]\[
f(-2) = \sqrt{-6(-2) + 19} = \sqrt{12 + 19} = \sqrt{31}
\][/tex]
Therefore:
[tex]\[
f(-2) = 5.5677643628300215
\][/tex]
2. Evaluating [tex]\( g(-3) \)[/tex]:
The function [tex]\( g(x) \)[/tex] is given by:
[tex]\[
g(x) = \frac{x}{x^2 + 15}
\][/tex]
Plugging [tex]\( x = -3 \)[/tex] into the function, we get:
[tex]\[
g(-3) = \frac{-3}{(-3)^2 + 15} = \frac{-3}{9 + 15} = \frac{-3}{24} = -\frac{1}{8}
\][/tex]
Simplified, we have:
[tex]\[
g(-3) = -0.125
\][/tex]
3. Evaluating [tex]\( h\left(-\frac{1}{4}\right) \)[/tex]:
The function [tex]\( h(x) \)[/tex] is given by:
[tex]\[
h(x) = -18 + |12x|
\][/tex]
Plugging [tex]\( x = -\frac{1}{4} \)[/tex] into the function, we get:
[tex]\[
h\left(-\frac{1}{4}\right) = -18 + \left|12 \left(-\frac{1}{4}\right)\right| = -18 + \left|-3\right| = -18 + 3 = -15
\][/tex]
Therefore:
[tex]\[
h\left(-\frac{1}{4}\right) = -15.0
\][/tex]
So the simplified answers are:
[tex]\[
\begin{array}{c}
f(-2) = 5.5677643628300215 \\
g(-3) = -0.125 \\
h\left(-\frac{1}{4}\right) = -15.0
\end{array}
\][/tex]
