Let's break down the given problem step by step.

We are given: $g(f(x)) = \frac{3x - 5}{2x - 2}, \quad x \neq 1$ and $f(x) = 3x - 1$

We are tasked with finding $g(-2)$.

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

We know that: $g(f(x)) = \frac{3x - 5}{2x - 2}$ But we need to find $g(-2)$, so we must first substitute $f(x)$ into this expression.

Step 2: Set $f(x) = 3x - 1$

From the function $f(x) = 3x - 1$, substitute $f(x) = y$, so we rewrite the given equation: $g(y) = \frac{3x - 5}{2x - 2}, \quad \text{where } y = f(x)$ To find $g(-2)$, we need to solve for the value of $x$ such that $f(x) = -2$.

Step 3: Solve for $x$ when $f(x) = -2$

We set $f(x) = 3x - 1$ equal to $-2$: $3x - 1 = -2$ Now, solve for $x$: $3x = -2 + 1$ $3x = -1$ $x = -\frac{1}{3}$

Step 4: Substitute $x = -\frac{1}{3}$ into $g(f(x))$

Now, substitute $x = -\frac{1}{3}$ into the expression for $g(f(x))$: $g(f(x)) = \frac{3x - 5}{2x - 2}$ Substitute $x = -\frac{1}{3}$ into this expression: $g(f(-\frac{1}{3})) = \frac{3\left(-\frac{1}{3}\right) - 5}{2\left(-\frac{1}{3}\right) - 2}$ Simplify the numerator and the denominator:

Numerator: $3\left(-\frac{1}{3}\right) - 5 = -1 - 5 = -6$

Denominator: $2\left(-\frac{1}{3}\right) - 2 = -\frac{2}{3} - 2 = -\frac{2}{3} - \frac{6}{3} = -\frac{8}{3}$

Step 5: Final computation

Now, the expression becomes: $g(-2) = \frac{-6}{-\frac{8}{3}} = -6 \times -\frac{3}{8} = \frac{18}{8} = \frac{9}{4}$

Thus, the value of $g(-2)$ is: $g(-2) = \frac{9}{4}$

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. What is the general formula for the composition of two functions $g(f(x))$?
2. How do you find the inverse of a linear function?
3. If $f(x) = 2x + 3$ and $g(x) = \frac{1}{x - 1}$, what is $g(f(1))$?
4. How would you graph the function $g(x) = \frac{3x - 5}{2x - 2}$?
5. How can you solve function composition problems using algebraic substitution?

Tip: Always check the domain restrictions in function problems to avoid undefined values!