Consider the function represented by [tex]\(9x + 3y = 12\)[/tex] with [tex]\(x\)[/tex] as the independent variable. How can this function be written using function notation?

A. [tex]\(f(v) = -\frac{1}{3} v + \frac{4}{3}\)[/tex]
B. [tex]\(f(x) = -3x + 4\)[/tex]
C. [tex]\(f(x) = -\frac{1}{3} x + \frac{4}{3}\)[/tex]
D. [tex]\(f(y) = -3y + 4\)[/tex]



Answer :

To write the function represented by the equation [tex]\( 9x + 3y = 12 \)[/tex] in function notation with [tex]\( x \)[/tex] as the independent variable, we need to isolate [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].

1. Start with the original equation:
[tex]\[ 9x + 3y = 12 \][/tex]

2. Subtract [tex]\( 9x \)[/tex] from both sides to isolate the [tex]\( 3y \)[/tex] term:
[tex]\[ 3y = 12 - 9x \][/tex]

3. Divide both sides by [tex]\( 3 \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{12 - 9x}{3} \][/tex]

4. Simplify the right-hand side:
[tex]\[ y = \frac{12}{3} - \frac{9x}{3} \][/tex]
[tex]\[ y = 4 - 3x \][/tex]

Now that we have [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], we can rewrite the equation as a function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = -3x + 4 \][/tex]

Therefore, the function in notation form is:
[tex]\[ f(x) = -3x + 4 \][/tex]

Among the given choices, the correct function notation is:
[tex]\[ f(x) = -3x + 4 \][/tex]