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please give the correct answer step by step

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!

Diketahui g∘f(x) = (3x - 5) / (2x - 2), x ≠ 1 dan f(x) = 3x - 1, maka nilai g(-2) = ....

Learn how to find the value of g(-2) for the given function composition where g∘f(x) = (3x - 5) / (2x - 2) and f(x) = 3x - 1. Step-by-step solution suitable for students in Grades 10-12, covering function composition and algebraic manipulation.

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