We are given two functions, $f(x) = 5x - 9$ and $g(x) = 6 - x^2$. We need to calculate the following: (a) $f(g(-5))$ (b) $g(f(-5))$ (c) $f(f(-5))$ (d) $g(g(-5))$

1. Problem Description

We are given two functions, $f(x) = 5x - 9$ and $g(x) = 6 - x^2$. We need to calculate the following:

(a) $f(g(-5))$

(b) $g(f(-5))$

(c) $f(f(-5))$

(d) $g(g(-5))$

2. Solution Steps

(a) $f(g(-5))$

First, we need to find $g(-5)$.

$g(-5) = 6 - (-5)^2 = 6 - 25 = -19$

Then, we need to find $f(g(-5)) = f(-19)$.

$f(-19) = 5(-19) - 9 = -95 - 9 = -104$

(b) $g(f(-5))$

First, we need to find $f(-5)$.

$f(-5) = 5(-5) - 9 = -25 - 9 = -34$

Then, we need to find $g(f(-5)) = g(-34)$.

$g(-34) = 6 - (-34)^2 = 6 - 1156 = -1150$

(c) $f(f(-5))$

We already found $f(-5)$ in part (b): $f(-5) = -34$.

So, we need to find $f(f(-5)) = f(-34)$.

$f(-34) = 5(-34) - 9 = -170 - 9 = -179$

(d) $g(g(-5))$

We already found $g(-5)$ in part (a): $g(-5) = -19$.

So, we need to find $g(g(-5)) = g(-19)$.

$g(-19) = 6 - (-19)^2 = 6 - 361 = -355$

3. Final Answer

(a) $f(g(-5)) = -104$

(b) $g(f(-5)) = -1150$

(c) $f(f(-5)) = -179$

(d) $g(g(-5)) = -355$