To find $f(g(x))$ and $g(f(x))$, we'll substitute each function into the other.

Given: $f(x) = \frac{1}{x} - 5 \quad \text{and} \quad g(x) = 9x + 5$

Step 1: Find $f(g(x))$

1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = f(9x + 5)$
2. Since $f(x) = \frac{1}{x} - 5$, replace $x$ in $f(x)$ with $9x + 5$: $f(g(x)) = \frac{1}{9x + 5} - 5$

So, $f(g(x)) = \frac{1}{9x + 5} - 5$

Step 2: Find $g(f(x))$

1. Substitute $f(x)$ into $g(x)$: $g(f(x)) = g\left(\frac{1}{x} - 5\right)$
2. Since $g(x) = 9x + 5$, replace $x$ in $g(x)$ with $\frac{1}{x} - 5$: $g(f(x)) = 9\left(\frac{1}{x} - 5\right) + 5$
3. Distribute $9$: $g(f(x)) = \frac{9}{x} - 45 + 5$
4. Combine like terms: $g(f(x)) = \frac{9}{x} - 40$

So, $g(f(x)) = \frac{9}{x} - 40$

Final Answers

$f(g(x)) = \frac{1}{9x + 5} - 5$ $g(f(x)) = \frac{9}{x} - 40$

Would you like further details or have any questions?

Related Questions

1. What are the domain restrictions for $f(g(x))$ and $g(f(x))$?
2. How would you find $g(g(x))$ or $f(f(x))$ using the same functions?
3. What are the inverses of $f(x)$ and $g(x)$, if they exist?
4. How would changes in the constants (like $-5$ or $+5$) affect the results?
5. Can we compose functions in the reverse order, and how does it affect the outcome?

Tip

When composing functions, keep in mind that $f(g(x))$ and $g(f(x))$ often have different domains based on the inner functions. Always check for any restrictions before simplifying.