Answered

The functions [tex]f, g[/tex], and [tex]h[/tex] are defined as follows:
[tex]\[ f(x) = \frac{x^2 + 3}{x + 5} \quad g(x) = |-18 + 8x| \quad h(x) = 5 + \sqrt{x - 3} \][/tex]

Find [tex]f(5), g\left(\frac{1}{4}\right)[/tex], and [tex]h(8)[/tex].
Simplify your answers as much as possible.

[tex]\[
\begin{aligned}
f(5) & = \square \\
g\left(\frac{1}{4}\right) & = \square \\
h(8) & = \square
\end{aligned}
\][/tex]



Answer :

GinnyAnswer
Let's evaluate each function step-by-step at the given points:

1. Evaluating [tex]\( f(5) \)[/tex]:

The function [tex]\( f \)[/tex] is given by:
[tex]\[ f(x) = \frac{x^2 + 3}{x + 5} \][/tex]

Plugging in [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{5^2 + 3}{5 + 5} \][/tex]
[tex]\[ f(5) = \frac{25 + 3}{10} \][/tex]
[tex]\[ f(5) = \frac{28}{10} \][/tex]
[tex]\[ f(5) = 2.8 \][/tex]

So,
[tex]\[ f(5) = 2.8 \][/tex]

2. Evaluating [tex]\( g\left(\frac{1}{4}\right) \)[/tex]:

The function [tex]\( g \)[/tex] is given by:
[tex]\[ g(x) = |-18 + 8x| \][/tex]

Plugging in [tex]\( x = \frac{1}{4} \)[/tex]:
[tex]\[ g\left(\frac{1}{4}\right) = |-18 + 8 \cdot \frac{1}{4}| \][/tex]
[tex]\[ g\left(\frac{1}{4}\right) = |-18 + 2| \][/tex]
[tex]\[ g\left(\frac{1}{4}\right) = |-16| \][/tex]
[tex]\[ g\left(\frac{1}{4}\right) = 16 \][/tex]

So,
[tex]\[ g\left(\frac{1}{4}\right) = 16 \][/tex]

3. Evaluating [tex]\( h(8) \)[/tex]:

The function [tex]\( h \)[/tex] is given by:
[tex]\[ h(x) = 5 + \sqrt{x - 3} \][/tex]

Plugging in [tex]\( x = 8 \)[/tex]:
[tex]\[ h(8) = 5 + \sqrt{8 - 3} \][/tex]
[tex]\[ h(8) = 5 + \sqrt{5} \][/tex]

To simplify further, we keep [tex]\( \sqrt{5} \)[/tex] as it is:
[tex]\[ h(8) \approx 5 + 2.23606797749979 \][/tex] (since [tex]\( \sqrt{5} \approx 2.23606797749979 \)[/tex])

So,
[tex]\[ h(8) \approx 7.23606797749979 \][/tex]

In summary:
[tex]\[ \begin{aligned} f(5) & = 2.8 \\ g\left(\frac{1}{4}\right) & = 16 \\ h(8) & \approx 7.23606797749979 \end{aligned} \][/tex]