https://file.aiquickdraw.com/mathbot/file/17350795280400m4cg4sh.jpg

No, the function in the image is not "1-1." It is given as:

\[
f(x) = 1 - 4x - 3e^{2x-1}.
\]

To determine if the function is **one-to-one (1-1)**, we need to verify if every value of \( f(x) \) corresponds to a unique value of \( x \). This is equivalent to testing whether the function is strictly increasing or decreasing, meaning the derivative \( f'(x) \) must not change sign (it should always be positive or always negative).

### Steps to check if \( f(x) \) is one-to-one:
1. **Compute the derivative** of \( f(x) \):
   \[
   f(x) = 1 - 4x - 3e^{2x-1}.
   \]
   Differentiating with respect to \( x \):
   \[
   f'(x) = -4 - 6e^{2x-1}.
   \]

2. **Analyze the derivative**:
   - \( e^{2x-1} \) is an exponential term and always positive for all real \( x \).
   - Therefore, \( -6e^{2x-1} \) is always negative.
   - Adding \( -4 \), the derivative becomes:
     \[
     f'(x) = -4 - 6e^{2x-1},
     \]
     which is strictly **negative** for all \( x \).

3. **Conclusion**:
   Since \( f'(x) < 0 \) for all \( x \), the function is strictly decreasing. This ensures that \( f(x) \) is one-to-one.

Thus, **yes, the function \( f(x) \) is one-to-one**.

Would you like a step-by-step explanation of the derivative or further clarification?

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### Related Questions:
1. What does it mean for a function to be one-to-one?
2. How do we test if a function is strictly increasing or decreasing?
3. Can a function with exponential terms like \( e^{2x-1} \) ever change direction?
4. What is the domain and range of this function?
5. How does the derivative relate to monotonicity?

**Tip:** A one-to-one function always passes the **horizontal line test**—no horizontal line intersects the graph more than once.

Determine whether the function f(x) = 1 - 4x - 3e^{2x-1} is one-to-one using calculus and monotonicity. This problem explores derivatives and exponential functions, providing a detailed step-by-step solution. Suitable for advanced high school and college students.

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