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The functions [tex]f, g[/tex], and [tex]h[/tex] are defined as follows:
[tex]\ \textless \ br/\ \textgreater \ f(x) = \sqrt{-6x + 19} \quad g(x) = \frac{x}{x^2 + 15} \quad h(x) = -18 + |12x|\ \textless \ br/\ \textgreater \ [/tex]

Find [tex]f(-2), g(-3)[/tex], and [tex]h\left(-\frac{1}{4}\right)[/tex].
Simplify your answers as much as possible.
[tex]\ \textless \ br/\ \textgreater \ \begin{array}{c}\ \textless \ br/\ \textgreater \ f(-2) = \square \\\ \textless \ br/\ \textgreater \ g(-3) = \square \\\ \textless \ br/\ \textgreater \ h\left(-\frac{1}{4}\right) = \square\ \textless \ br/\ \textgreater \ \end{array}\ \textless \ br/\ \textgreater \ [/tex]



Answer :

GinnyAnswer
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]

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