To find the range of the function $(f + g)(x) = 3x^2 - 2 + \frac{1}{5}x^2 + 4$, we need to simplify the expression first and then analyze the resulting function.

Step-by-Step Solution:

1. Combine the functions:

   Let's simplify the expression for $(f + g)(x)$:

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

2. Combine like terms:

   Combine the $x^2$ terms and the constant terms:

   $(f + g)(x) = \left(3 + \frac{1}{5}\right)x^2 + (-2 + 4).$

   Let's simplify the coefficients:

   $(f + g)(x) = \left(3 + \frac{1}{5}\right)x^2 + 2.$

3. Convert to a common fraction:

   Convert $3$ to a fraction with a denominator of $5$:

   $3 = \frac{15}{5}.$

   So, we have:

   $3 + \frac{1}{5} = \frac{15}{5} + \frac{1}{5} = \frac{16}{5}.$

   Now, rewrite the function:

   $(f + g)(x) = \frac{16}{5}x^2 + 2.$

4. Analyze the quadratic function:

   The function $(f + g)(x) = \frac{16}{5}x^2 + 2$ is a quadratic function in the standard form $ax^2 + bx + c$, where $a = \frac{16}{5} > 0$, $b = 0$, and $c = 2$.

   Since the coefficient of $x^2$ ($a = \frac{16}{5}$) is positive, the parabola opens upwards.

5. Determine the range of the function:

   • For a quadratic function that opens upwards, the minimum value occurs at the vertex of the parabola.
   • Since there is no $x$ term ($b = 0$), the vertex is at $x = 0$.

   Plug $x = 0$ into the function to find the minimum value:

   $(f + g)(0) = \frac{16}{5} \cdot 0^2 + 2 = 2.$

   The minimum value of $(f + g)(x)$ is $2$.

6. Range of the function:

   • The function can take on any value greater than or equal to $2$ because the parabola opens upwards and has a minimum value at $2$.

Conclusion:

The range of the function $(f + g)(x) = \frac{16}{5}x^2 + 2$ is:

$\text{Range} = [2, \infty).$

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

Here are 5 related questions that expand on this information:

1. How do you find the vertex of a quadratic function in standard form?
2. What are the effects of the coefficient $a$ in the quadratic function $ax^2 + bx + c$ on the graph?
3. How can you determine the range of a quadratic function that opens downwards?
4. What transformations occur when you add or subtract a constant to a function?
5. How do you graph the quadratic function $(f + g)(x) = \frac{16}{5}x^2 + 2$?

Tip: To determine the range of a quadratic function, always look at the direction in which the parabola opens (upwards or downwards) and find the vertex, which represents the minimum or maximum value of the function.