https://file.aiquickdraw.com/mathbot/file/1727532762224nb95sag8.jpg

The problem provides several functions:

- \( a(x) = x - 5 \)
- \( b(x) = 3x \)
- \( c(x) = -\frac{x}{2} \)
- \( d(x) = \sqrt{x} \)

You are asked to determine the values of the following expressions:

- \( a(b(c(d(x)))) \)
- \( a(b(a(x))) \)
- \( a(a(a(x))) \)
- \( d(d(d(x))) \)

Let's calculate each option one by one:

### a. \( a(b(c(d(x)))) \)
1. Start with \( d(x) = \sqrt{x} \).
2. Then \( c(d(x)) = c(\sqrt{x}) = -\frac{\sqrt{x}}{2} \).
3. Next, \( b(c(d(x))) = b\left(-\frac{\sqrt{x}}{2}\right) = 3\left(-\frac{\sqrt{x}}{2}\right) = -\frac{3\sqrt{x}}{2} \).
4. Finally, \( a(b(c(d(x)))) = a\left(-\frac{3\sqrt{x}}{2}\right) = \left(-\frac{3\sqrt{x}}{2}\right) - 5 \).

So, the result for option a is \( -\frac{3\sqrt{x}}{2} - 5 \).

### b. \( a(b(a(x))) \)
1. First, \( a(x) = x - 5 \).
2. Then, \( b(a(x)) = b(x - 5) = 3(x - 5) = 3x - 15 \).
3. Finally, \( a(b(a(x))) = a(3x - 15) = (3x - 15) - 5 = 3x - 20 \).

So, the result for option b is \( 3x - 20 \).

### c. \( a(a(a(x))) \)
1. Start with \( a(x) = x - 5 \).
2. Then \( a(a(x)) = a(x - 5) = (x - 5) - 5 = x - 10 \).
3. Finally, \( a(a(a(x))) = a(x - 10) = (x - 10) - 5 = x - 15 \).

So, the result for option c is \( x - 15 \).

### d. \( d(d(d(x))) \)
1. Start with \( d(x) = \sqrt{x} \).
2. Then \( d(d(x)) = d(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{\frac{1}{4}} \).
3. Finally, \( d(d(d(x))) = d(x^{\frac{1}{4}}) = \sqrt{x^{\frac{1}{4}}} = x^{\frac{1}{8}} \).

So, the result for option d is \( x^{\frac{1}{8}} \).

### Summary of results:
- a. \( -\frac{3\sqrt{x}}{2} - 5 \)
- b. \( 3x - 20 \)
- c. \( x - 15 \)
- d. \( x^{\frac{1}{8}} \)

Let me know if you'd like further details or have any questions!

Here are 5 related questions you might want to explore:
1. How can we simplify compositions of more complex functions?
2. How would the results change if we modified the functions \( a(x) \), \( b(x) \), \( c(x) \), or \( d(x) \)?
3. What is the general approach for calculating nested functions?
4. How can you differentiate a composition of functions like \( a(b(c(d(x)))) \)?
5. How does the order of function composition affect the result?

**Tip:** When composing functions, always start from the innermost function and work outward to avoid confusion!

Diketahui bahwa a(x) = x - 5, b(x) = 3x, c(x) = -x/2, dan d(x) = sqrt(x). Pilihlah jawaban yang benar untuk a(b(c(d(x)))) atau varian lainnya.

Explore how to solve function compositions involving algebraic functions a(x) = x - 5, b(x) = 3x, c(x) = -x/2, and d(x) = sqrt(x). This problem involves calculating nested functions and is suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation