Hodges Brewer
09/07/2023 · Primary School
Composition of a function with itself Suppose that the functions \( f \) and \( g \) are defined as follows. \( f(x)=x^{2}+4 \) \( g(x)=\frac{2}{5 x}, x \neq 0 \) Find the compositions \( f \) of and \( g \) og. Simplify your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition. You do not have to indicate the domain.
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the compositions \( f(g(x)) \) and \( g(f(x)) \), we will substitute one function into the other.
1. **Finding \( f(g(x)) \)**:
We start with the function \( g(x) = \frac{2}{5x} \). We will substitute this into \( f(x) \).
\[
f(g(x)) = f\left(\frac{2}{5x}\right)
\]
Now, we substitute \( \frac{2}{5x} \) into the function \( f(x) = x^2 + 4 \):
\[
f\left(\frac{2}{5x}\right) = \left(\frac{2}{5x}\right)^2 + 4
\]
Calculating \( \left(\frac{2}{5x}\right)^2 \):
\[
\left(\frac{2}{5x}\right)^2 = \frac{4}{25x^2}
\]
Now, substituting this back into the expression for \( f(g(x)) \):
\[
f(g(x)) = \frac{4}{25x^2} + 4
\]
To combine the terms, we can express \( 4 \) as \( \frac{100}{25} \):
\[
f(g(x)) = \frac{4}{25x^2} + \frac{100}{25} = \frac{4 + 100x^2}{25x^2}
\]
Thus, we have:
\[
f(g(x)) = \frac{4 + 100x^2}{25x^2}
\]
2. **Finding \( g(f(x)) \)**:
Now we will substitute \( f(x) = x^2 + 4 \) into \( g(x) \):
\[
g(f(x)) = g(x^2 + 4)
\]
Substituting \( x^2 + 4 \) into \( g(x) = \frac{2}{5x} \):
\[
g(x^2 + 4) = \frac{2}{5(x^2 + 4)}
\]
Therefore, we have:
\[
g(f(x)) = \frac{2}{5(x^2 + 4)}
\]
In summary, the compositions are:
\[
f(g(x)) = \frac{4 + 100x^2}{25x^2}
\]
\[
g(f(x)) = \frac{2}{5(x^2 + 4)}
\]
Quick Answer
\[
f(g(x)) = \frac{4 + 100x^2}{25x^2}
\]
\[
g(f(x)) = \frac{2}{5(x^2 + 4)}
\]
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