We are given two functions:

• $f(x) = 5x + 6$
• $g(x) = -3$

The questions are:

(a) $f(g(4))$ (b) $g(f(-5))$

Let's solve them step by step.

(a) $f(g(4))$

1. First, we find $g(4)$. Since $g(x) = -3$ for all $x$, we know that: $g(4) = -3$
2. Next, we substitute $g(4) = -3$ into $f(x)$: $f(-3) = 5(-3) + 6 = -15 + 6 = -9$

Thus, $f(g(4)) = -9$.

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

1. First, we find $f(-5)$: $f(-5) = 5(-5) + 6 = -25 + 6 = -19$
2. Next, we substitute $f(-5) = -19$ into $g(x)$. Since $g(x) = -3$ for all $x$, we know that: $g(-19) = -3$

Thus, $g(f(-5)) = -3$.

Final Answers:

• $f(g(4)) = -9$
• $g(f(-5)) = -3$

Would you like further details or have any questions about the steps?

Here are some related questions:

1. What is the general definition of composite functions?
2. How would you approach calculating $f(g(x))$ for a new pair of functions?
3. Can a constant function like $g(x) = -3$ ever be injective or surjective?
4. What happens if $f(x)$ and $g(x)$ are both linear functions?
5. How would you graph $f(x) = 5x + 6$ and $g(x) = -3$?

Tip: Always start solving composite functions from the inside and work your way outward!