To find which expression represents the function with [tex]\( x \)[/tex] as the independent variable, we start with the given equation:
[tex]\[ y - 6x - 9 = 0 \][/tex]
We need to rewrite this in a form where [tex]\( y \)[/tex] is expressed in terms of [tex]\( x \)[/tex]. This can be identified as the slope-intercept form of a linear equation, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here are the steps:
1. Isolate [tex]\( y \)[/tex]:
[tex]\[ y - 6x - 9 = 0 \][/tex]
[tex]\[ y = 6x + 9 \][/tex]
This represents the function with [tex]\( x \)[/tex] as the independent variable. In function notation, this can be written as:
[tex]\[ f(x) = 6x + 9 \][/tex]
Now we match this with the given options:
- [tex]\( f(x) = \frac{1}{6}x + \frac{3}{2} \)[/tex]: This does not match our equation.
- [tex]\( f(n) = 6 y + 9 \)[/tex]: This uses inappropriate variables and does not match the form.
- [tex]\( f(n) = \frac{1}{6}v + \frac{3}{2} \)[/tex]: This uses inappropriate variables and does not match the form.
The correct function notation with [tex]\( x \)[/tex] as the independent variable is:
[tex]\[ f(x) = 6x + 9 \][/tex]
Therefore, the correct answer is:
[tex]\[ f(x)=6x+9 \][/tex]
