Let's tackle each part of this problem step-by-step.

Problem Overview

1. Given function $f(x) = \frac{2}{3x - 5}$
   • Part (a): Find $f\left(\frac{1}{3}\right)$
   • Part (b): Find the inverse function $f^{-1}(x)$
2. Given function $g(x) = 5x^2 - 20x + 23$
   • Part (c): Express $g(x)$ in the form $a(x - b)^2 + c$

Solution

Part (a): Find $f\left(\frac{1}{3}\right)$

To find $f\left(\frac{1}{3}\right)$, substitute $x = \frac{1}{3}$ into $f(x)$:

$f\left(\frac{1}{3}\right) = \frac{2}{3\left(\frac{1}{3}\right) - 5} = \frac{2}{1 - 5} = \frac{2}{-4} = -\frac{1}{2}$

So, $f\left(\frac{1}{3}\right) = -\frac{1}{2}$

Part (b): Find $f^{-1}(x)$

To find the inverse function $f^{-1}(x)$, start by setting $y = f(x)$ and solving for $x$ in terms of $y$:

1. Let $y = \frac{2}{3x - 5}$.
2. Multiply both sides by $3x - 5$: $y(3x - 5) = 2$
3. Expand and rearrange to isolate $x$: $3xy - 5y = 2$ $3xy = 2 + 5y$ $x = \frac{2 + 5y}{3y}$

Thus, the inverse function is: $f^{-1}(x) = \frac{2 + 5x}{3x}$

Part (c): Express $g(x) = 5x^2 - 20x + 23$ in the form $a(x - b)^2 + c$

To rewrite $g(x) = 5x^2 - 20x + 23$ in the form $a(x - b)^2 + c$, complete the square.

1. Factor out the coefficient of $x^2$ from the first two terms: $g(x) = 5(x^2 - 4x) + 23$
2. Complete the square inside the parentheses:
   • Take half of $-4$, square it to get $4$, and add/subtract $4$ inside the parentheses. $g(x) = 5(x^2 - 4x + 4 - 4) + 23$ $= 5((x - 2)^2 - 4) + 23$ $= 5(x - 2)^2 - 20 + 23$ $= 5(x - 2)^2 + 3$

So, in the form $a(x - b)^2 + c$, we have: $g(x) = 5(x - 2)^2 + 3$

Summary of Answers

1. (a) $f\left(\frac{1}{3}\right) = -\frac{1}{2}$
2. (b) $f^{-1}(x) = \frac{2 + 5x}{3x}$
3. (c) $g(x) = 5(x - 2)^2 + 3$

Would you like further details on any part, or have any additional questions?

Follow-Up Questions

1. What are the general steps to finding the inverse of a function?
2. How does completing the square help in converting a quadratic function to vertex form?
3. Could the domain of $f(x) = \frac{2}{3x - 5}$ affect the inverse function?
4. Why is vertex form useful for analyzing the graph of a quadratic function?
5. How would you verify that $f^{-1}(f(x)) = x$ for the inverse found?

Tip

When finding inverses, always check if the function is one-to-one in its domain to ensure that an inverse exists.