We are given three functions: $f(x) = -9x$, $g(x) = 7x + 3$, and $h(x) = 6x^2 + 7x$. We need to evaluate the following composite functions: $f(g(x))$, $g(f(x))$, $g(g(x))$, and $h(f(x))$.

1. Problem Description

We are given three functions: $f(x) = -9x$, $g(x) = 7x + 3$, and $h(x) = 6x^2 + 7x$. We need to evaluate the following composite functions: $f(g(x))$, $g(f(x))$, $g(g(x))$, and $h(f(x))$.

2. Solution Steps

a) Evaluate $f(g(x))$:

We need to substitute $g(x)$ into $f(x)$.

$f(g(x)) = f(7x+3) = -9(7x+3)$

$f(g(x)) = -63x - 27$

b) Evaluate $g(f(x))$:

We need to substitute $f(x)$ into $g(x)$.

$g(f(x)) = g(-9x) = 7(-9x) + 3$

$g(f(x)) = -63x + 3$

c) Evaluate $g(g(x))$:

We need to substitute $g(x)$ into $g(x)$.

$g(g(x)) = g(7x+3) = 7(7x+3) + 3$

$g(g(x)) = 49x + 21 + 3$

$g(g(x)) = 49x + 24$

d) Evaluate $h(f(x))$:

We need to substitute $f(x)$ into $h(x)$.

$h(f(x)) = h(-9x) = 6(-9x)^2 + 7(-9x)$

$h(f(x)) = 6(81x^2) - 63x$

$h(f(x)) = 486x^2 - 63x$

3. Final Answer

$f(g(x)) = -63x - 27$

$g(f(x)) = -63x + 3$

$g(g(x)) = 49x + 24$

$h(f(x)) = 486x^2 - 63x$