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

To determine the domain of \( f(x) + g(x) \), we first need to understand the domains of \( f(x) \) and \( g(x) \) individually.

Given:
- \( f(x) = 3x^2 - 2 \)
- \( g(x) = \frac{1}{5}x^2 + 4 \)

### Domain of \( f(x) \):
The function \( f(x) = 3x^2 - 2 \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of \( f(x) \) is all real numbers:
\[ \text{Domain of } f(x) = (-\infty, \infty) \]

### Domain of \( g(x) \):
The function \( g(x) = \frac{1}{5}x^2 + 4 \) is also a polynomial (since it is a quadratic function). Polynomials are defined for all real numbers as well. So, the domain of \( g(x) \) is:
\[ \text{Domain of } g(x) = (-\infty, \infty) \]

### Domain of \( f(x) + g(x) \):
Since both \( f(x) \) and \( g(x) \) are defined for all real numbers, the domain of their sum \( f(x) + g(x) \) is the intersection of their domains. Since both domains are \( (-\infty, \infty) \), the domain of \( f(x) + g(x) \) is also:
\[ \text{Domain of } f(x) + g(x) = (-\infty, \infty) \]

This means that \( f(x) + g(x) \) is defined for all real numbers.

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you determine the domain of a rational function?
2. What is the range of \( f(x) + g(x) \) given the functions above?
3. How would you find the domain of a composite function like \( f(g(x)) \)?
4. What are the general rules for finding the domain of a function involving square roots?
5. How do you find the domain of a function that includes both polynomials and square roots?

**Tip:** The domain of a function is the set of all possible input values (x-values) that will not cause any undefined behavior in the function, such as division by zero or taking the square root of a negative number.

Understand how to determine the domain of f(x) + g(x) where f(x) = 3x^2 - 2 and g(x) = 1/5x^2 + 4. Learn about polynomial functions, their domains, and how to find the domain of their sum. Suitable for high school students studying algebra and functions.

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