To determine the correct function notation for the equation [tex]\(9x + 3y = 12\)[/tex] with [tex]\(y\)[/tex] as the dependent variable and [tex]\(x\)[/tex] as the independent variable, we need to rearrange the equation to solve for [tex]\(y\)[/tex]. Here’s a step-by-step solution:
1. Start with the given equation:
[tex]\[ 9x + 3y = 12 \][/tex]
2. Isolate [tex]\(y\)[/tex] on one side of the equation:
[tex]\[ 3y = 12 - 9x \][/tex]
3. Solve for [tex]\(y\)[/tex] by dividing every term by 3:
[tex]\[ y = \frac{12 - 9x}{3} \][/tex]
4. Simplify the fraction:
[tex]\[ y = \frac{12}{3} - \frac{9x}{3} \][/tex]
[tex]\[ y = 4 - 3x \][/tex]
5. In function notation, we represent [tex]\(y\)[/tex] as [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 4 - 3x \][/tex]
Now, compare this function with the given options:
1. [tex]\( f(v) = -\frac{1}{3}v + \frac{4}{3} \)[/tex]
2. [tex]\( f(x) = -3x + 4 \)[/tex]
3. [tex]\( f(x) = -\frac{1}{3}x + \frac{4}{3} \)[/tex]
4. [tex]\( f(t) = -3v + 4 \)[/tex]
We can see that the correctly rearranged function [tex]\(f(x) = 4 - 3x\)[/tex] matches with option 2 when we reorder the terms:
[tex]\[ f(x) = -3x + 4 \][/tex]
Thus, the correct function notation for the given equation is:
[tex]\[ f(x) = -3x + 4 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{2} \][/tex]
