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

To find the domain of the function \(f + g\), we first need to understand the domains of the individual functions \(f(x)\) and \(g(x)\).

### Functions and Their Domains

1. **Function \(f(x) = 3x^2 - 2\)**

   - This is a quadratic function. 
   - Quadratic functions are defined for all real numbers.
   - So, the domain of \(f(x)\) is all real numbers: \(\mathbb{R}\).

2. **Function \(g(x) = \frac{1}{5x^2 - 4}\)**

   - This is a rational function.
   - A rational function is undefined when its denominator is equal to zero. 
   - Therefore, we need to find the values of \(x\) that make the denominator zero:

\[
5x^2 - 4 = 0
\]

\[
5x^2 = 4
\]

\[
x^2 = \frac{4}{5}
\]

\[
x = \pm \sqrt{\frac{4}{5}} = \pm \frac{2}{\sqrt{5}} = \pm \frac{2\sqrt{5}}{5}
\]

- The function \(g(x)\) is undefined when \(x = \frac{2\sqrt{5}}{5}\) or \(x = -\frac{2\sqrt{5}}{5}\).

- Thus, the domain of \(g(x)\) is all real numbers except \(x = \pm \frac{2\sqrt{5}}{5}\).

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

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

- Since \(f(x)\) is defined for all real numbers and \(g(x)\) is defined for all real numbers except \(x = \pm \frac{2\sqrt{5}}{5}\), the domain of \(f + g\) will also be all real numbers except where \(g(x)\) is undefined.

\[
\text{Domain of } f+g = \mathbb{R} \setminus \left\{ \pm \frac{2\sqrt{5}}{5} \right\}
\]

### Conclusion

The domain of the function \(f + g\) is:

\[
\boxed{\{ x \in \mathbb{R} : x \neq \frac{2\sqrt{5}}{5}, x \neq -\frac{2\sqrt{5}}{5} \}}
\]

Would you like more details or have any questions?

### Relative Questions

1. What is the range of \(f(x) = 3x^2 - 2\)?
2. How do we find the range of a rational function like \(g(x) = \frac{1}{5x^2 - 4}\)?
3. How can we graph \(f(x)\) and \(g(x)\) to visualize their domains and ranges?
4. What happens to the function \(f+g\) at the points where \(g(x)\) is undefined?
5. What is the derivative of \(f(x) + g(x)\)?

### Tip

When finding the domain of a function, always check where the function might be undefined, such as division by zero in rational functions or taking the square root of a negative number in real functions.

Explore how to determine the domain of the function f + g where f(x) = 3x^2 - 2 and g(x) = 1/(5x^2 - 4). This detailed analysis covers quadratic and rational functions, examining their domains and intersections. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation