Long Howell
03/15/2024 · High School
Suppose \( H(x)=(6 x-5)^{5} \). Find two functions \( f \) and \( g \) such that \( (f \circ g)(x)=H(x) \). Neither function can be the identity function. (There may be more than one correct answer.)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find two functions \( f \) and \( g \) such that \( (f \circ g)(x) = H(x) \), we need to express \( H(x) \) as a composition of two functions.
Given:
\[ H(x) = (6x - 5)^5 \]
Let's express \( H(x) \) as a composition of two functions \( f \) and \( g \) such that \( (f \circ g)(x) = H(x) \).
We can express \( H(x) \) as \( f(g(x)) \), where \( f(x) \) and \( g(x) \) are the two functions we are looking for.
Let's choose \( g(x) = 6x - 5 \) and \( f(x) = x^5 \).
Now, we can verify if \( (f \circ g)(x) = H(x) \) by substituting the functions into the composition.
\[ (f \circ g)(x) = f(g(x)) = f(6x - 5) = (6x - 5)^5 \]
Since \( (f \circ g)(x) = H(x) \), we have found two functions \( f(x) = x^5 \) and \( g(x) = 6x - 5 \) such that \( (f \circ g)(x) = H(x) \).
Therefore, the two functions are:
\[ f(x) = x^5 \]
\[ g(x) = 6x - 5 \]
Quick Answer
The two functions are \( f(x) = x^5 \) and \( g(x) = 6x - 5 \).
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit