The problem provided in the image asks for the result of translating the function $f(x) = 2(3x - 5) + 1$ by a translation vector $\left( \frac{-2}{3} \right)$.

Step-by-step explanation:

1. Original function:
   $f(x) = 2(3x - 5) + 1$ Simplifying it: $f(x) = 6x - 10 + 1 = 6x - 9$ So, the function is now $f(x) = 6x - 9$.

2. Translation vector:
   The translation vector $\left( \frac{-2}{3} \right)$ suggests a translation in both the $x$-direction and the $y$-direction. The exact form is unclear from the image, but generally, a translation applies changes to both the $x$ and $y$ coordinates.

3. Impact on the function: A translation affects the function based on how the vector modifies $x$ and $y$. If we are translating the input $x$-value by $\frac{-2}{3}$, this affects $x$ in the following way:

   $f(x)$ is transformed to $f\left( x - \frac{-2}{3} \right) = f\left( x + \frac{2}{3} \right)$.

4. Applying the translation: We substitute $x + \frac{2}{3}$ into the function: $f\left( x + \frac{2}{3} \right) = 6\left( x + \frac{2}{3} \right) - 9$ Simplifying the expression: $= 6x + 4 - 9 = 6x - 5$

So, after the translation, the new function becomes $f(x) = 6x - 5$.

Final Answer:

The image of the function $f(x)$ under the translation by $\left( \frac{-2}{3} \right)$ is $f(x) = 6x - 5$.

Do you want further details or have any questions?

Here are some related questions to expand on the topic:

1. How do different translations affect linear functions like this one?
2. What happens if the translation vector affects both $x$ and $y$?
3. Can we apply translations to non-linear functions similarly?
4. How does a translation in the $y$-direction only affect a function?
5. What are the general rules for translating functions geometrically?

Tip: When applying translations, be careful to adjust the input $x$-value in accordance with the vector components.