Let's solve the two parts of the problem step by step.

Given:

• $f(x) = 6x - 5$
• $g(x) = 7 - x^2$

(a) $f(g(0))$

1. First, calculate $g(0)$: $g(0) = 7 - (0)^2 = 7$

2. Now, substitute $g(0) = 7$ into $f(x)$: $f(g(0)) = f(7) = 6(7) - 5 = 42 - 5 = 37$

So, $f(g(0)) = 37$.

(b) $g(f(0))$

1. First, calculate $f(0)$: $f(0) = 6(0) - 5 = -5$

2. Now, substitute $f(0) = -5$ into $g(x)$: $g(f(0)) = g(-5) = 7 - (-5)^2 = 7 - 25 = -18$

So, $g(f(0)) = -18$.

Final Answers:

• (a) $f(g(0)) = 37$
• (b) $g(f(0)) = -18$

Would you like more details on any step?

Here are 5 related questions you might find interesting:

1. What is the general process for calculating the composition of two functions?
2. How would you solve $g(f(x))$ if given a different function for $f(x)$?
3. What happens if you reverse the order of function composition, such as calculating $f(g(x))$ instead of $g(f(x))$?
4. Can you graph the functions $f(x) = 6x - 5$ and $g(x) = 7 - x^2$ to visualize the results?
5. What are the domain and range for the composite function $f(g(x))$?

Tip: When working with composite functions, always solve from the inside out, starting with the innermost function.