when f(x)=3x²-2 and g(x)=1/5x²=4 what is the domain of f+g(X)

To determine the domain of \(f + g(x)\) when \(f(x) = 3x^2 - 2\) and \(g(x) = \frac{1}{5}x^2 + 4\), we need to first analyze the domains of each individual function, \(f(x)\) and \(g(x)\), and then combine them.

### Step-by-Step Analysis:

1. **Domain of \(f(x) = 3x^2 - 2\):**

   - The function \(f(x) = 3x^2 - 2\) is a quadratic function.
   - Quadratic functions are defined for all real values of \(x\).
   - Therefore, the domain of \(f(x)\) is all real numbers, \( \mathbb{R} \).

2. **Domain of \(g(x) = \frac{1}{5}x^2 + 4\):**

   - The function \(g(x) = \frac{1}{5}x^2 + 4\) is also a quadratic function.
   - Quadratic functions are defined for all real values of \(x\).
   - Therefore, the domain of \(g(x)\) is all real numbers, \( \mathbb{R} \).

3. **Domain of \(f + g(x) = f(x) + g(x)\):**

   - To find the domain of \(f + g(x)\), we take the intersection of the domains of \(f(x)\) and \(g(x)\).
   - Since both \(f(x)\) and \(g(x)\) are defined for all real numbers, the intersection is also all real numbers.

### Conclusion:

The domain of \(f + g(x) = 3x^2 - 2 + \frac{1}{5}x^2 + 4\) is all real numbers, \( \mathbb{R} \).

### Domain of \(f + g(x)\):

\[
\text{Domain of } (f + g)(x) = \mathbb{R}.
\]

Would you like more details, or do you have any questions?

Here are 5 related questions that expand on this information:

1. What is the range of the function \(f + g(x) = 3x^2 - 2 + \frac{1}{5}x^2 + 4\)?
2. How do you find the domain of a function that involves a square root?
3. What happens to the domain of a function when a denominator is introduced?
4. Can a quadratic function ever have a restricted domain? Under what circumstances?
5. How do transformations like translations and reflections affect the domain of a function?

**Tip:** Always remember that the domain of a function is the set of all input values (x-values) for which the function is defined.

Explore the domain of the function f + g(x) = 3x^2 - 2 + (1/5)x^2 + 4. Understand how to determine the domain by analyzing the individual domains of f(x) = 3x^2 - 2 and g(x) = (1/5)x^2 + 4, and combining them. Suitable for high school students learning about quadratic functions and domain analysis.

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