Let's solve the problem step-by-step. We're asked to find [tex]\((f \circ g)(4)\)[/tex] for the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], and then round the answer to two decimal places if necessary.
The functions given are:
[tex]\[ f(x) = \sqrt{\frac{5x}{8}} \][/tex]
[tex]\[ g(x) = 1 + \sqrt{x} \][/tex]
We start by evaluating [tex]\(g(4)\)[/tex]:
1. Evaluate [tex]\(g(4)\)[/tex]:
[tex]\[ g(x) = 1 + \sqrt{x} \][/tex]
So, for [tex]\(x = 4\)[/tex]:
[tex]\[ g(4) = 1 + \sqrt{4} \][/tex]
[tex]\[ g(4) = 1 + 2 = 3 \][/tex]
Next, we use this result as the input for the function [tex]\(f(x)\)[/tex]:
2. Evaluate [tex]\(f(g(4))\)[/tex]:
We have already found [tex]\(g(4) = 3\)[/tex], so we need to evaluate [tex]\(f(3)\)[/tex]:
[tex]\[ f(x) = \sqrt{\frac{5x}{8}} \][/tex]
For [tex]\(x = 3\)[/tex]:
[tex]\[ f(3) = \sqrt{\frac{5 \cdot 3}{8}} = \sqrt{\frac{15}{8}} \][/tex]
3. Compute [tex]\(\sqrt{\frac{15}{8}}\)[/tex]:
[tex]\[ \sqrt{\frac{15}{8}} \approx 1.3693063937629153 \][/tex]
Finally, we need to round this result to two decimal places:
4. Round the result to two decimal places:
[tex]\[ 1.3693063937629153 \approx 1.37 \][/tex]
Therefore, [tex]\((f \circ g)(4) \)[/tex] is approximately [tex]\(1.37\)[/tex].
