For the given functions f and​ g, complete parts​ (a)-(h). For parts​ (a)-(d), also find the domain.
f left parenthesis x right parenthesis equals StartRoot 5 x EndRoot​; g left parenthesis x right parenthesis equals 9 x minus 5
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Part 1
​(a) Find ​(fplus​g)(x).
​(fplus​g)(x)equals

enter your response here
​ (Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

To solve the problem, we'll first need to determine the functions \( f(x) \) and \( g(x) \) and then find the sum \( (f + g)(x) \).

Given:
\[
f(x) = \sqrt{5x}
\]
\[
g(x) = 9x - 5
\]

### Step 1: Find \( (f + g)(x) \)
The sum of two functions \( f(x) \) and \( g(x) \) is defined as:
\[
(f + g)(x) = f(x) + g(x)
\]

So, we have:
\[
(f + g)(x) = \sqrt{5x} + (9x - 5)
\]

This is the simplified expression for \( (f + g)(x) \).

### Step 2: Find the Domain
The domain of \( (f + g)(x) \) is the set of all \( x \) values for which both \( f(x) \) and \( g(x) \) are defined.

- For \( f(x) = \sqrt{5x} \), the expression under the square root must be non-negative. Thus:
  \[
  5x \geq 0
  \]
  \[
  x \geq 0
  \]
  So, the domain of \( f(x) \) is \( x \geq 0 \).

- For \( g(x) = 9x - 5 \), it's a linear function, so its domain is all real numbers, \( x \in \mathbb{R} \).

The domain of \( (f + g)(x) \) is the intersection of the domains of \( f(x) \) and \( g(x) \), which is \( x \geq 0 \).

### Final Answer:
\[
(f + g)(x) = \sqrt{5x} + 9x - 5 \quad \text{with domain} \quad x \geq 0
\]

Would you like further details or explanations? Here are some additional questions related to this topic:

1. How do you find the domain of a function involving a square root?
2. What is the process to add two functions together?
3. How does the domain of \( f(x) \) affect the domain of \( (f+g)(x) \)?
4. Can \( (f+g)(x) \) ever be a linear function? Why or why not?
5. How would the problem change if \( g(x) \) included a square root?

**Tip:** When dealing with square roots in functions, always check for non-negative values inside the root to determine the domain.

Explore the sum of functions f(x) = √(5x) and g(x) = 9x - 5. Learn how to find (f + g)(x), including simplification and determining the domain. Suitable for high school students studying functions and domain analysis.

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