Let's solve this step-by-step.

1. Given Functions:

   • $f(x) = x^2 + 8x + 5$
   • $g(x) = \frac{3}{x^2 - 2}$

2. Find $f(x - 3)$: Substitute $x - 3$ into $f(x)$: $f(x - 3) = (x - 3)^2 + 8(x - 3) + 5$

   Expanding this:

   • $(x - 3)^2 = x^2 - 6x + 9$
   • $8(x - 3) = 8x - 24$

   So: $f(x - 3) = x^2 - 6x + 9 + 8x - 24 + 5$ Simplify by combining like terms: $f(x - 3) = x^2 + 2x - 10$

3. Find $g\left(\frac{5}{x}\right)$: Substitute $\frac{5}{x}$ into $g(x)$: $g\left(\frac{5}{x}\right) = \frac{3}{\left(\frac{5}{x}\right)^2 - 2}$

   Simplifying the square of $\frac{5}{x}$:

   • $\left(\frac{5}{x}\right)^2 = \frac{25}{x^2}$

   So: $g\left(\frac{5}{x}\right) = \frac{3}{\frac{25}{x^2} - 2}$

   To combine the terms in the denominator, express $-2$ with a common denominator: $g\left(\frac{5}{x}\right) = \frac{3}{\frac{25 - 2x^2}{x^2}}$

   Simplify by inverting the fraction in the denominator: $g\left(\frac{5}{x}\right) = \frac{3x^2}{25 - 2x^2}$

Final Answers:

• $f(x - 3) = x^2 + 2x - 10$
• $g\left(\frac{5}{x}\right) = \frac{3x^2}{25 - 2x^2}$

Would you like more details on the steps, or do you have any questions about the simplifications?

Related Questions:

1. How would you find $f(x + 2)$ using the same method?
2. What would $g\left(\frac{x}{2}\right)$ evaluate to?
3. How can we verify the result of $f(x - 3)$ by substitution?
4. What other substitutions can we make in $g(x)$ to explore different function values?
5. How can derivatives of $f(x)$ and $g(x)$ be used in applications?

Tip:

When substituting values in functions, always simplify step-by-step to avoid errors in combining like terms.