https://file.aiquickdraw.com/mathbot/file/172809697703718zudwdk.jpg

Let's analyze the problem and solve it.

The given equation is:
\[
4f(x) + 2f(-x) = 5x + 3
\]
where \( f(x) \) is a linear function.

1. We know that a linear function can be written as \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

2. Let’s substitute \( f(x) = ax + b \) and \( f(-x) = -ax + b \) into the equation:
\[
4(ax + b) + 2(-ax + b) = 5x + 3
\]
Expanding both sides:
\[
4(ax + b) = 4ax + 4b
\]
\[
2(-ax + b) = -2ax + 2b
\]
Now combine the terms:
\[
(4ax - 2ax) + (4b + 2b) = 5x + 3
\]
Simplify:
\[
2ax + 6b = 5x + 3
\]
By comparing the coefficients of \( x \) and the constants, we get:
\[
2a = 5 \quad \text{and} \quad 6b = 3
\]
Solving these equations:
\[
a = \frac{5}{2}, \quad b = \frac{1}{2}
\]

Thus, the function is:
\[
f(x) = \frac{5}{2}x + \frac{1}{2}
\]

### Now, let's check the statements:

1. \( f(1) = -3 \):
   \[
   f(1) = \frac{5}{2}(1) + \frac{1}{2} = \frac{5}{2} + \frac{1}{2} = 3 \quad (\text{not } -3)
   \]
   **False**

2. \( f^{-1}(8) = 3 \): To find the inverse of \( f(x) = \frac{5}{2}x + \frac{1}{2} \), we solve for \( x \):
   \[
   y = \frac{5}{2}x + \frac{1}{2}
   \]
   Solve for \( x \):
   \[
   2y = 5x + 1 \quad \Rightarrow \quad 5x = 2y - 1 \quad \Rightarrow \quad x = \frac{2y - 1}{5}
   \]
   The inverse function is:
   \[
   f^{-1}(y) = \frac{2y - 1}{5}
   \]
   Now, substitute \( y = 8 \):
   \[
   f^{-1}(8) = \frac{2(8) - 1}{5} = \frac{16 - 1}{5} = \frac{15}{5} = 3
   \]
   **True**

3. \( f(-3) = 1 \):
   \[
   f(-3) = \frac{5}{2}(-3) + \frac{1}{2} = -\frac{15}{2} + \frac{1}{2} = -\frac{14}{2} = -7
   \]
   **False**

4. \( f^{-1}(-2) = -1 \):
   Substitute \( y = -2 \) into the inverse function:
   \[
   f^{-1}(-2) = \frac{2(-2) - 1}{5} = \frac{-4 - 1}{5} = \frac{-5}{5} = -1
   \]
   **True**

### Conclusion:
Statements 2 and 4 are true. Therefore, the correct answer is **C**: 2 and 4 SAJA yang benar.

Would you like any further details or clarifications? Here are five related questions:

1. How is the inverse function of a linear function generally derived?
2. What does it mean for a function to be linear?
3. What is the significance of the slope \( a \) in the linear function \( f(x) = ax + b \)?
4. How does the inverse function reflect the original function graphically?
5. How can you verify if a function has an inverse?

**Tip**: Always check the inverse function by plugging the result back into the original function.

Diketahui 4f(x) + 2f(-x) = 5x + 3 dengan f(x) merupakan fungsi linear. Berdasarkan informasi tersebut, manakah di antara pilihan berikut yang bernilai benar? 1. f(1) = -3 2. f^{-1}(8) = 3 3. f(-3) = 1 4. f^{-1}(-2) = -1

Learn how to solve a linear function equation involving both f(x) and f(-x). This problem involves verifying conditions of a linear function and its inverse. Suitable for students in Grades 10-12.

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