The problem provided asks to find the values of various function compositions based on the given functions $f(x) = x^2 - 1$ and $g(x) = 2x - 1$. Let's solve each part:

1. $f(3)$: $f(3) = 3^2 - 1 = 9 - 1 = 8$

2. $g(3)$: $g(3) = 2(3) - 1 = 6 - 1 = 5$

3. $g(1)$: $g(1) = 2(1) - 1 = 2 - 1 = 1$

4. $f(1)$: $f(1) = 1^2 - 1 = 1 - 1 = 0$

5. $g(f(3))$: First, find $f(3)$: $f(3) = 8$ Then, substitute into $g(x)$: $g(8) = 2(8) - 1 = 16 - 1 = 15$

6. $f(g(3))$: First, find $g(3)$: $g(3) = 5$ Then, substitute into $f(x)$: $f(5) = 5^2 - 1 = 25 - 1 = 24$

7. $f(g(1))$: First, find $g(1)$: $g(1) = 1$ Then, substitute into $f(x)$: $f(1) = 0$

8. $g(f(1))$: First, find $f(1)$: $f(1) = 0$ Then, substitute into $g(x)$: $g(0) = 2(0) - 1 = 0 - 1 = -1$

Final answers:

• (i) $f(3) = 8$
• (ii) $g(3) = 5$
• (iii) $g(1) = 1$
• (iv) $f(1) = 0$
• (v) $g(f(3)) = 15$
• (vi) $f(g(3)) = 24$
• (vii) $f(g(1)) = 0$
• (viii) $g(f(1)) = -1$

Would you like further details or explanations for any step?

Related questions:

1. How do you compose two functions in general?
2. What are the domain and range of $f(x)$ and $g(x)$?
3. What is the significance of finding compositions of functions?
4. How do you interpret the result when one function is substituted into another?
5. Can the order of function composition change the result?

Tip: Always compute the inner function first when evaluating composite functions.