Find an equation of the line. write the equation using function notation. through (4,−7); perpendicular to 3y=x−6

To make an line perpendicular to another, you must take the slope of the first line and find its negative inverse to put in the second.
3y=x-6 --> y=(x-6)/3 --> y=1/3x-2
The slope of the first line is 1/3 so the slope of the second line will be -3.
Now you must use the point slope formula to find the actual equation of the second line.
y-y1=m(x-x1) --> y-(-7)=-3(x-4) --> y+7=-3x+12 --> y=-3x+5
Now you need to translate that to function notation, which is pretty easy. Simply remember f(x) = y. So...
f(x)=-3x+5, and that's your answer.
Remember to check your answers using a graphing calculator. If you don't have one, you can use one online. I recommend using https://www.desmos.com/calculator.