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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.

Evaluate (f ∘ g)(-3) and (g ∘ f)(1) where f(x) = 5x - 3 and g(x) = √(x² - 1).

Learn how to evaluate composite functions with the problem (f ∘ g)(-3) and (g ∘ f)(1), where f(x) = 5x - 3 and g(x) = √(x² - 1). This problem covers key concepts such as function composition and square root evaluation. Suitable for high school students in Grades 10-12.

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