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Part 1

If 

​f(v)equals=StartFraction 1 Over v squared plus 2 EndFraction1v2+2 

and 

​g(v)equals=StartRoot v plus 5 EndRootv+5​, 

find 

​(fcircle◦​g)(v) 

and 

​(gcircle◦​f)(v).

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Part 1

​(fcircle◦​g)(v)equals=enter your response here 

​(Simplify your​ answer.)

Part 2

​(gcircle◦​f)(v)equals=enter your response here 

​(Simplify your​ answer.)

Let's solve the problem step by step.

### Given Functions:
\[
f(v) = \frac{1}{v^2 + 2}
\]
\[
g(v) = \sqrt{v + 5}
\]

### Part 1: Find \((f \circ g)(v)\)

The composition \((f \circ g)(v)\) means that we substitute \( g(v) \) into \( f(v) \). That is, \( (f \circ g)(v) = f(g(v)) \).

Substituting \( g(v) = \sqrt{v + 5} \) into \( f(v) \):

\[
f(g(v)) = \frac{1}{(\sqrt{v + 5})^2 + 2}
\]

Simplify the expression:

\[
(\sqrt{v + 5})^2 = v + 5
\]

Thus,

\[
f(g(v)) = \frac{1}{(v + 5) + 2} = \frac{1}{v + 7}
\]

So, 

\[
(f \circ g)(v) = \frac{1}{v + 7}
\]

### Part 2: Find \((g \circ f)(v)\)

The composition \((g \circ f)(v)\) means that we substitute \( f(v) \) into \( g(v) \). That is, \( (g \circ f)(v) = g(f(v)) \).

Substituting \( f(v) = \frac{1}{v^2 + 2} \) into \( g(v) \):

\[
g(f(v)) = \sqrt{\frac{1}{v^2 + 2} + 5}
\]

To simplify further:

\[
g(f(v)) = \sqrt{\frac{1 + 5(v^2 + 2)}{v^2 + 2}} = \sqrt{\frac{1 + 5v^2 + 10}{v^2 + 2}}
\]

Simplify the numerator:

\[
g(f(v)) = \sqrt{\frac{5v^2 + 11}{v^2 + 2}}
\]

So,

\[
(g \circ f)(v) = \sqrt{\frac{5v^2 + 11}{v^2 + 2}}
\]

### Final Answers:

- \((f \circ g)(v) = \frac{1}{v + 7}\)
- \((g \circ f)(v) = \sqrt{\frac{5v^2 + 11}{v^2 + 2}}\)

Would you like further details on these steps or to explore related problems?

Here are 5 related questions to consider:
1. How would \((f \circ g)(v)\) and \((g \circ f)(v)\) change if the functions \( f(v) \) or \( g(v) \) were altered slightly?
2. What is the domain of \((f \circ g)(v)\) and \((g \circ f)(v)\)?
3. How can you verify that your compositions are correct?
4. What would happen if you added a constant to both \( f(v) \) and \( g(v) \)?
5. How could you extend these compositions to three or more functions?

**Tip:** When simplifying functions, ensure that you correctly apply arithmetic operations, particularly when dealing with square roots and powers.

Explore how to compute (f ◦ g)(v) and (g ◦ f)(v) with step-by-step solutions. Understand the concepts of function composition and apply them to specific functions involving square roots and fractions. Suitable for advanced high school students.

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