https://file.aiquickdraw.com/mathbot/file/1728886362922gzf34i91.jpg

We are asked to find the compositions \( (f \circ f)(x) \) and \( (g \circ g)(x) \), where the functions are defined as follows:

\[
f(x) = \frac{7}{5x}, \quad x \neq 0
\]
\[
g(x) = 2x - 9
\]

### 1. Finding \( (f \circ f)(x) \)
This means we substitute \( f(x) \) into itself:

\[
(f \circ f)(x) = f(f(x)) = f\left(\frac{7}{5x}\right)
\]

Now, using the definition of \( f(x) \):

\[
f\left(\frac{7}{5x}\right) = \frac{7}{5 \left( \frac{7}{5x} \right)} = \frac{7}{\frac{35}{5x}} = \frac{7 \cdot 5x}{35} = \frac{35x}{35} = x
\]

So,

\[
(f \circ f)(x) = x
\]

### 2. Finding \( (g \circ g)(x) \)
Next, we compute \( g(g(x)) \):

\[
(g \circ g)(x) = g(g(x)) = g(2x - 9)
\]

Using the definition of \( g(x) \):

\[
g(2x - 9) = 2(2x - 9) - 9 = 4x - 18 - 9 = 4x - 27
\]

So,

\[
(g \circ g)(x) = 4x - 27
\]

### Final Answers:
- \( (f \circ f)(x) = x \)
- \( (g \circ g)(x) = 4x - 27 \)

Would you like more details or explanations on any part? Here are five related questions to explore further:

1. What happens when composing functions \( f \circ g \) or \( g \circ f \)?
2. How do the domains of composite functions affect the final result?
3. What does it mean geometrically when \( f \circ f(x) = x \)?
4. Can we compose functions more than twice, such as \( (f \circ f \circ f)(x) \)?
5. How would we approach compositions with more complicated functions, like quadratics or rationals?

### Tip: 
When composing functions, always substitute carefully, keeping track of all terms, especially in cases involving fractions or more complex expressions!

Find the compositions (f ∘ f)(x) and (g ∘ g)(x) given that f(x) = 7/(5x), x ≠ 0 and g(x) = 2x - 9. Simplify your answers.

Learn how to find and simplify the compositions of functions (f ∘ f)(x) and (g ∘ g)(x) for f(x) = 7/(5x), x ≠ 0, and g(x) = 2x - 9. This problem covers key algebraic concepts, including function composition and simplification of rational expressions. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation