Let me analyze the contents of the image and provide a solution to the questions.

Summary of the problem:

1) Given:

• $f(x) = 2x^2 - 3$
• $g(x) = 4 + 7x$

Find:

• $f(g(x))$
• $g(f(x))$
• $g(g(x))$
• $f(f(x))$

2) Given:

• $t(x) = 4x - 5$
• $g(x) = 7 - 20x$

Find:

• $f(g(x))$
• $f(g(-1))$
• $g(f(x))$
• $g(f(-1))$
• $f(f(t))$
• $g(g(t))$

I'll now compute these step by step. sympy import symbols, Function, simplify

Define the variables and functions for Question 1

x = symbols('x') f = Function('f')(x) g = Function('g')(x) f_expr = 2x**2 - 3 g_expr = 4 + 7x

Composite functions for Question 1

f_g = f_expr.subs(x, g_expr) # f(g(x)) g_f = g_expr.subs(x, f_expr) # g(f(x)) g_g = g_expr.subs(x, g_expr) # g(g(x)) f_f = f_expr.subs(x, f_expr) # f(f(x))

Define the variables and functions for Question 2

t = Function('t')(x) t_expr = 4x - 5 g_expr_2 = 7 - 20x

Composite functions for Question 2

f_g_2 = f_expr.subs(x, g_expr_2) # f(g(x)) f_g_minus_1 = f_g_2.subs(x, -1) # f(g(-1)) g_f_2 = g_expr_2.subs(x, f_expr) # g(f(x)) g_f_minus_1 = g_f_2.subs(x, -1) # g(f(-1)) f_f_t = f_expr.subs(x, t_expr) # f(f(t)) g_g_t = g_expr_2.subs(x, t_expr) # g(g(t))

Simplify all results

results = { "f(g(x))": simplify(f_g), "g(f(x))": simplify(g_f), "g(g(x))": simplify(g_g), "f(f(x))": simplify(f_f), "f(g(-1))": simplify(f_g_minus_1), "g(f(-1))": simplify(g_f_minus_1), "f(f(t))": simplify(f_f_t), "g(g(t))": simplify(g_g_t) }

resultsHere are the solutions to the problems:

Question 1:

1. $f(g(x)) = 2(7x + 4)^2 - 3$
2. $g(f(x)) = 14x^2 - 17$
3. $g(g(x)) = 49x + 32$
4. $f(f(x)) = 2(2x^2 - 3)^2 - 3$

Question 2:

1. $f(g(x)) = 2(7 - 20x)^2 - 3$
2. $f(g(-1)) = 1455$
3. $g(f(x)) = 107 - 80x$
4. $g(f(-1)) = 27$
5. $f(f(t)) = 2(4x - 5)^2 - 3$
6. $g(g(t)) = 107 - 80x$

Would you like detailed steps for any of these computations?

Here are 5 follow-up questions to enhance understanding:

1. What are the general rules for computing composite functions like $f(g(x))$?
2. How does substitution work when finding $g(g(x))$ or $f(f(x))$?
3. Why is simplification important in composite functions?
4. How can you verify the results of composite functions graphically?
5. How do the results change when substituting specific values, like $g(-1)$?

Tip: Always simplify expressions at each step to make the final results clearer and easier to interpret.