If f(x)=x2+3x and g(x)=10x, find g(f(n)).

Given the functions \( f(x) = x^2 + 3x \) and \( g(x) = 10x \), we need to find \( g(f(n)) \).

### Step 1: Substitute \( f(n) \) into \( g(x) \).
We know that \( f(n) = n^2 + 3n \).

So, \( g(f(n)) = g(n^2 + 3n) \).

### Step 2: Apply the function \( g(x) \).
The function \( g(x) \) is defined as \( g(x) = 10x \).

Therefore, \( g(n^2 + 3n) = 10 \times (n^2 + 3n) \).

### Step 3: Distribute the multiplication.
\[
g(n^2 + 3n) = 10n^2 + 30n
\]

### Final Answer:
\[
g(f(n)) = 10n^2 + 30n
\]

Would you like further explanation on any part of this process? Here are some related questions you might consider:

1. How do you find \( f(g(n)) \) if \( g(x) = 10x \) and \( f(x) = x^2 + 3x \)?
2. What is the domain of \( g(f(n)) \)?
3. How would the answer change if \( f(x) = x^2 - 3x \)?
4. What is the significance of the composition \( g(f(n)) \) in real-world applications?
5. How would you find the inverse of \( g(f(n)) \)?

**Tip:** When working with polynomial functions in compositions, it’s often helpful to expand and simplify the expressions fully to see the result clearly.

Discover how to compute g(f(n)) using functions f(x) = x^2 + 3x and g(x) = 10x. This problem involves understanding function composition, polynomial functions, and basic algebraic operations. Suitable for students in Grades 10-12.

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