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!