Evaluating the Composition of Functions q and r

Oct 13, 2025 by miodeya.com 68 views

Introduction

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. In this article, we will evaluate the composition of functions q and r, and write the domain in interval notation. The functions q and r are defined as follows:

q(x) = 1 / (x - 8)
r(x) = |6x - 7|

Evaluating the Composition of Functions q and r

To evaluate the composition of functions q and r, we need to substitute the expression for r(x) into the function q(x). This means that we will replace x in the function q(x) with the expression |6x - 7|.

(q ∘ r)(x) = q(r(x)) = q(|6x - 7|) = 1 / (|6x - 7| - 8)

Simplifying the Expression

To simplify the expression, we need to find a common denominator for the terms in the denominator. However, since the absolute value function is involved, we need to consider two cases:

Case 1: 6x - 7 ≥ 0

In this case, the absolute value function can be rewritten as:

|6x - 7| = 6x - 7

Substituting this into the expression, we get:

(q ∘ r)(x) = 1 / ((6x - 7) - 8) = 1 / (6x - 15)

Case 2: 6x - 7 < 0

In this case, the absolute value function can be rewritten as:

|6x - 7| = -(6x - 7)

Substituting this into the expression, we get:

(q ∘ r)(x) = 1 / (-(6x - 7) - 8) = 1 / (-6x + 15)

Writing the Domain in Interval Notation

To write the domain in interval notation, we need to consider the restrictions on the domain imposed by the functions q and r.

For the function q(x) = 1 / (x - 8), the domain is all real numbers except x = 8.

For the function r(x) = |6x - 7|, the domain is all real numbers.

However, since the composition of functions q and r involves the absolute value function, we need to consider the restrictions imposed by the absolute value function.

In Case 1, where 6x - 7 ≥ 0, the domain is all real numbers except x = 15/6.

In Case 2, where 6x - 7 < 0, the domain is all real numbers except x = 15/6.

Conclusion

In conclusion, the composition of functions q and r is given by:

(q ∘ r)(x) = 1 / (6x - 15) for 6x - 7 ≥ 0

(q ∘ r)(x) = 1 / (-6x + 15) for 6x - 7 < 0

The domain of the composition of functions q and r is all real numbers except x = 15/6.

Final Answer

The final answer is:

(q ∘ r)(x) = 1 / (6x - 15) for 6x - 7 ≥ 0

(q ∘ r)(x) = 1 / (-6x + 15) for 6x - 7 < 0

Domain: (-∞, 15/6) ∪ (15/6, ∞)

Note: The final answer is in interval notation, where (-∞, 15/6) ∪ (15/6, ∞) represents all real numbers except x = 15/6.

Introduction

In our previous article, we evaluated the composition of functions q and r, and wrote the domain in interval notation. In this article, we will answer some frequently asked questions about the composition of functions q and r.

Q: What is the composition of functions q and r?

A: The composition of functions q and r is given by:

(q ∘ r)(x) = q(r(x)) = q(|6x - 7|) = 1 / (|6x - 7| - 8)

Q: How do I simplify the expression for the composition of functions q and r?

A: To simplify the expression, we need to find a common denominator for the terms in the denominator. However, since the absolute value function is involved, we need to consider two cases:

Case 1: 6x - 7 ≥ 0

In this case, the absolute value function can be rewritten as:

|6x - 7| = 6x - 7

Substituting this into the expression, we get:

(q ∘ r)(x) = 1 / ((6x - 7) - 8) = 1 / (6x - 15)

Case 2: 6x - 7 < 0

In this case, the absolute value function can be rewritten as:

|6x - 7| = -(6x - 7)

Substituting this into the expression, we get:

(q ∘ r)(x) = 1 / (-(6x - 7) - 8) = 1 / (-6x + 15)

Q: What is the domain of the composition of functions q and r?

A: The domain of the composition of functions q and r is all real numbers except x = 15/6.

Q: How do I write the domain in interval notation?

A: To write the domain in interval notation, we need to consider the restrictions on the domain imposed by the functions q and r.

In Case 1, where 6x - 7 ≥ 0, the domain is all real numbers except x = 15/6.

In Case 2, where 6x - 7 < 0, the domain is all real numbers except x = 15/6.

The domain in interval notation is:

(-∞, 15/6) ∪ (15/6, ∞)

Q: What is the final answer for the composition of functions q and r?

A: The final answer is:

(q ∘ r)(x) = 1 / (6x - 15) for 6x - 7 ≥ 0

(q ∘ r)(x) = 1 / (-6x + 15) for 6x - 7 < 0

Domain: (-∞, 15/6) ∪ (15/6, ∞)

Q: Can I use the composition of functions q and r in real-world applications?

A: Yes, the composition of functions q and r can be used in real-world applications such as:

Modeling population growth and decline
Analyzing financial data
Predicting weather patterns

Q: How do I apply the composition of functions q and r in real-world applications?

A: To apply the composition of functions q and r in real-world applications, you need to:

Identify the input and output variables
Substitute the input variables into the function
Evaluate the function using the given values
Interpret the results in the context of the real-world application

Conclusion

In conclusion, the composition of functions q and r is a powerful tool that can be used to model and analyze complex systems. By understanding the composition of functions q and r, you can apply it in real-world applications such as modeling population growth and decline, analyzing financial data, and predicting weather patterns.

Final Answer

The final answer is:

(q ∘ r)(x) = 1 / (6x - 15) for 6x - 7 ≥ 0

(q ∘ r)(x) = 1 / (-6x + 15) for 6x - 7 < 0

Domain: (-∞, 15/6) ∪ (15/6, ∞)