What is the range of the function f+g(x)=3x2−2+15x2+4f+g(x)=3x2−2+51​x2+4

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.

Learn how to find the range of the quadratic function f+g(x)=3x^2−2+15x^2+4 with a step-by-step solution. This problem covers the vertex form of a quadratic function and the determination of its range. Suitable for students in Grades 10-12.

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