We are given the function $f(x) = 2x - \frac{7}{6}$ and asked to find $f\left( -\frac{2}{3} \right)$.

To solve this, substitute $x = -\frac{2}{3}$ into the function:

$f\left( -\frac{2}{3} \right) = 2 \left( -\frac{2}{3} \right) - \frac{7}{6}$

First, calculate $2 \left( -\frac{2}{3} \right)$:

$2 \left( -\frac{2}{3} \right) = -\frac{4}{3}$

Now substitute this back into the equation:

$f\left( -\frac{2}{3} \right) = -\frac{4}{3} - \frac{7}{6}$

To subtract these fractions, we need a common denominator. The least common denominator (LCD) between 3 and 6 is 6. Rewrite $-\frac{4}{3}$ as:

$-\frac{4}{3} = -\frac{8}{6}$

Now, subtract:

$-\frac{8}{6} - \frac{7}{6} = -\frac{15}{6}$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

$-\frac{15}{6} = -\frac{5}{2}$

Thus, the value of $f\left( -\frac{2}{3} \right)$ is:

$f\left( -\frac{2}{3} \right) = -\frac{5}{2}$

Would you like any further explanation or clarification on this? Here are some related questions you might find interesting:

1. How would the value of $f(x)$ change if the function was $f(x) = 3x - \frac{7}{6}$?
2. What would happen if the fraction in the function's expression was different, like $f(x) = 2x + \frac{7}{3}$?
3. How can we graph the function $f(x) = 2x - \frac{7}{6}$?
4. What is the domain and range of the function $f(x) = 2x - \frac{7}{6}$?
5. How do we apply the same steps to find $f \left( \frac{1}{2} \right)$?

Tip: Always simplify fractions to their lowest terms to make calculations easier and more accurate!