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



Answer :

To express the given equation [tex]\(9x + 3y = 12\)[/tex] in function notation with [tex]\(x\)[/tex] as the independent variable, follow these steps:

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

2. Isolate [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
- Subtract [tex]\(9x\)[/tex] from both sides of the equation:
[tex]\[ 3y = 12 - 9x \][/tex]

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

- Simplify the expression on the right-hand side:
[tex]\[ y = 4 - 3x \][/tex]

3. Rewrite the equation in function notation:
Since [tex]\(y = 4 - 3x\)[/tex], we can express this as:
[tex]\[ f(x) = -3x + 4 \][/tex]

Therefore, the function in function notation with [tex]\(x\)[/tex] as the independent variable is [tex]\( f(x) = -3x + 4 \)[/tex].

Now, comparing the options given:
- [tex]\(f(y) = -\frac{1}{3} y + \frac{4}{3}\)[/tex]
- [tex]\(f(x) = -3x + 4\)[/tex]
- [tex]\(f(x) = -\frac{1}{3} x + \frac{4}{3}\)[/tex]

The correct function notation is:
[tex]\[ f(x) = -3 x + 4 \][/tex]