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]
