Sure, let's work through the problem step-by-step.
Given the function [tex]\( f(x) = 3x - \frac{9}{5} \)[/tex], we need to find the value of the expression [tex]\(\frac{f(a) - f(b)}{a - b}\)[/tex].
First, let's determine [tex]\( f(a) \)[/tex] and [tex]\( f(b) \)[/tex].
For [tex]\( f(a) \)[/tex]:
[tex]\[ f(a) = 3a - \frac{9}{5} \][/tex]
For [tex]\( f(b) \)[/tex]:
[tex]\[ f(b) = 3b - \frac{9}{5} \][/tex]
Next, we compute the numerator [tex]\( f(a) - f(b) \)[/tex]:
[tex]\[ f(a) - f(b) = \left( 3a - \frac{9}{5} \right) - \left( 3b - \frac{9}{5} \right) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ f(a) - f(b) = 3a - \frac{9}{5} - 3b + \frac{9}{5} \][/tex]
Observe that [tex]\(\frac{9}{5}\)[/tex] cancels out:
[tex]\[ f(a) - f(b) = 3a - 3b \][/tex]
Now, consider the denominator [tex]\( a - b \)[/tex]:
[tex]\[ a - b \][/tex]
Putting it all together, we calculate the expression [tex]\(\frac{f(a) - f(b)}{a - b}\)[/tex]:
[tex]\[ \frac{f(a) - f(b)}{a - b} = \frac{3a - 3b}{a - b} \][/tex]
Factor out the common term in the numerator:
[tex]\[ \frac{3a - 3b}{a - b} = \frac{3(a - b)}{a - b} \][/tex]
Notice the [tex]\( a - b \)[/tex] terms cancel each other out:
[tex]\[ \frac{3(a - b)}{a - b} = 3 \][/tex]
Thus, the value of the expression [tex]\(\frac{f(a) - f(b)}{a - b}\)[/tex] is:
[tex]\[ 3 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
