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?

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.