We are asked to evaluate $(f \circ g)(-3)$ and $(g \circ f)(1)$, where:

$f(x) = 5x - 3$ $g(x) = \sqrt{x^2 - 1}$

Step 1: Evaluate $(f \circ g)(-3)$

This means we first evaluate $g(-3)$ and then substitute the result into $f(x)$.

1. Calculate $g(-3)$: $g(-3) = \sqrt{(-3)^2 - 1} = \sqrt{9 - 1} = \sqrt{8}$

2. Now, substitute $g(-3) = \sqrt{8}$ into $f(x)$: $f(\sqrt{8}) = 5(\sqrt{8}) - 3 = 5\sqrt{8} - 3$

Thus, $(f \circ g)(-3) = 5\sqrt{8} - 3$.

Step 2: Evaluate $(g \circ f)(1)$

This means we first evaluate $f(1)$ and then substitute the result into $g(x)$.

1. Calculate $f(1)$: $f(1) = 5(1) - 3 = 5 - 3 = 2$

2. Now, substitute $f(1) = 2$ into $g(x)$: $g(2) = \sqrt{2^2 - 1} = \sqrt{4 - 1} = \sqrt{3}$

Thus, $(g \circ f)(1) = \sqrt{3}$.

Final Results:

$(f \circ g)(-3) = 5\sqrt{8} - 3$ $(g \circ f)(1) = \sqrt{3}$

Would you like further clarification on any of these steps?

Here are five related questions for further exploration:

1. How do you find the domain of composite functions like $f \circ g$?
2. Can you graph $f(x) = 5x - 3$ and $g(x) = \sqrt{x^2 - 1}$?
3. What is the inverse of the function $g(x) = \sqrt{x^2 - 1}$?
4. How do you determine the range of a composite function?
5. What happens if $x = 1$ for $(f \circ g)(x)$?

Tip: Always check the domain restrictions when dealing with square roots to ensure you're only working with real numbers.