Are these 2 seperate questions? If so...
1)
We have :
y = 1/3 x +5
From this, we know that the slope of the line is 1/3 (slope is defined as y variation over x variation, and for each time x increases by 1, y increases by 1/3)
We can therefore say that the slope of a line perpendicular to this one will be 3. (If 2 lines are perpendicular, their slopes are inverses so slope = 1 / (1/3) = 3 )
We therefore have :
y = 3 x + b
To find b, we use the fact that this line intercepts x at 2. This means it goes through point x = 2, y = 0. This is also expressed as (2, 0). If we put these values in the equation...
0 = 3 * 2 + b
0 = 6 + b
b = -6
We have that the equation of the line which we were asked to find is :
y = 3 x = 6.
2)
When they say f(x) - b(x) here's what they want : (I used D(x) in a totally arbitrary manner)
D(x) = f(x) - b(x)
Then all we do is substitute in the values of f(x) and b(x) and simplify...
D(x) = ( 2x^2 - x ) - ( 3x + 6 )
D(x) = 2x^2 - x - 3x + 6
D(x) = 2x^2 - 4x + 6
D(x) = x^2 - 2x + 3
If they ask for f( f(x) ) then here's how that works...
f(x) basically means f is a function of x. However, x doesn't need to be an integer. x can be anything, I could set x as being 4*u if I wanted to, all I'd do is substitute x = 4u into f(x) so I'd get something like this :
f(4u) = 2(4u)^2 - 4u
f(4u) = 32u^2 - 4u
So, f(f(x))... let's rewrite that as f(F(x)) just so you know which f/F I'm talking about
They basically want me to substitute x=F(x) into f(x) so we get :
f(f(x)) = 2(2x^2 - x)^2 - (2x^2 - x)
All I've done here is swap out the x's out of 2x^2 - x and put (2x^2 - x) instead. Then, we simplify...
f(f(x)) = 2( 4x^4 - 4x^3 + x^2) - 2x^2 - x
f(f(x)) = 8x^4 - 4x^3 + 2x^2 - 2x^2 - x
f(f(x)) = 8x^4 - 4x^3 - x
f(f(x)) = x * ( 8x^3 - 4x^2 - 1 )
