Given functions are:
First, determine the expression for \( f(g(x)) \). By substituting \( g(x) \) into \( f(x) \):
\(f(g(x)) = (x + 3)^2 - 7(x + 3)\)
Expand the terms:
\(f(g(x)) = x^2 + 6x + 9 - 7x - 21\)
Simplify the expression:
\(f(g(x)) = x^2 - x - 12\)
Next, consider the expression \( f(g(x)) - 3x \):
\(f(g(x)) - 3x = x^2 - x - 12 - 3x\)
Simplify further:
\(f(g(x)) - 3x = x^2 - 4x - 12\)
To find the minimum value of this expression, calculate its derivative and set it to zero.
The derivative of \( f(g(x)) - 3x \) is:
\(\frac{d}{dx} (x^2 - 4x - 12) = 2x - 4\)
Set the derivative to zero to find the critical point:
\(2x - 4 = 0\)
Solve for \( x \):
\(x = 2\)
Substitute \( x = 2 \) into the expression \( f(g(x)) - 3x \) to find the minimum value:
\(f(g(2)) - 3(2) = 2^2 - 4(2) - 12 = 4 - 8 - 12 = -16\)
The minimum value of \( f(g(x)) - 3x \) is -16.