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Let's work through the problem step by step.

We are given the following functions:

\[
a(x) = 4x, \quad b(x) = x^2, \quad c(x) = x - 5, \quad d(x) = \sqrt{x}
\]

The questions ask us to find the compositions of these functions.

### a. \( f(x) = (a \circ b \circ c \circ d)(x) \)

This means we first apply \( d(x) \), then \( c(x) \), then \( b(x) \), and finally \( a(x) \). Let's find \( f(x) \) step-by-step:

1. \( d(x) = \sqrt{x} \)
2. \( c(d(x)) = c(\sqrt{x}) = \sqrt{x} - 5 \)
3. \( b(c(d(x))) = b(\sqrt{x} - 5) = (\sqrt{x} - 5)^2 = x - 10\sqrt{x} + 25 \)
4. \( a(b(c(d(x)))) = a(x - 10\sqrt{x} + 25) = 4(x - 10\sqrt{x} + 25) = 4x - 40\sqrt{x} + 100 \)

Thus, \( f(x) = 4x - 40\sqrt{x} + 100 \).

### b. \( g(x) = (a \circ a \circ d)(x) \)

This means we first apply \( d(x) \), then apply \( a(x) \) twice. Let's find \( g(x) \):

1. \( d(x) = \sqrt{x} \)
2. \( a(d(x)) = a(\sqrt{x}) = 4\sqrt{x} \)
3. \( a(a(d(x))) = a(4\sqrt{x}) = 4(4\sqrt{x}) = 16\sqrt{x} \)

Thus, \( g(x) = 16\sqrt{x} \).

### c. \( g(x) = (b \circ c \circ b \circ c)(x) \)

This means we apply \( c(x) \), then \( b(x) \), then \( c(x) \), and finally \( b(x) \). Let's find \( g(x) \):

1. \( c(x) = x - 5 \)
2. \( b(c(x)) = b(x - 5) = (x - 5)^2 = x^2 - 10x + 25 \)
3. \( c(b(c(x))) = c(x^2 - 10x + 25) = x^2 - 10x + 25 - 5 = x^2 - 10x + 20 \)
4. \( b(c(b(c(x)))) = b(x^2 - 10x + 20) = (x^2 - 10x + 20)^2 \)

Thus, \( g(x) = (x^2 - 10x + 20)^2 \).

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Would you like more detailed explanations or have any other questions?

### Related Questions:
1. How do we calculate the composition of two or more functions?
2. Can the order of composition change the result?
3. How can we simplify complicated functions involving radicals and polynomials?
4. How does the chain rule apply to function composition in calculus?
5. What are the key differences between function composition and multiplication?

**Tip:** Always start function composition from the innermost function and work outward to avoid mistakes.

Diketahui persamaan-persamaan berikut:
 a(x) = 4x;
 b(x) = x^2;
 c(x) = x - 5;
 d(x) = √x;
 Tentukanlah:
 a. f(x) = a ∘ b ∘ c ∘ d(x)
 b. g(x) = a ∘ a ∘ d(x)
 c. g(x) = b ∘ c ∘ b ∘ c(x)

Learn how to solve function composition problems with a detailed solution. This question covers the composition of functions a(x) = 4x, b(x) = x^2, c(x) = x - 5, and d(x) = √x. Suitable for high school students (Grades 10-12).

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