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give me step by step answer in detail from the beginning

The problem in the image asks to find \((g \circ f)^{-1}(x)\), where the functions \( g(x) \) and \( f(x) \) are defined as follows:

- \( g(x) = 3x^2 - 3x - 1 \)
- \( f(x) = 4x - 6 \)

The composition \((g \circ f)(x)\) means \( g(f(x)) \), and the task is to find the inverse of this composition, \((g \circ f)^{-1}(x)\).

### Step-by-Step Solution:

#### Step 1: Find the composition \( (g \circ f)(x) = g(f(x)) \)
Substitute \( f(x) = 4x - 6 \) into the function \( g(x) = 3x^2 - 3x - 1 \).

\[
g(f(x)) = g(4x - 6)
\]

Substitute \( 4x - 6 \) into the formula for \( g(x) \):

\[
g(4x - 6) = 3(4x - 6)^2 - 3(4x - 6) - 1
\]

#### Step 2: Simplify \( g(4x - 6) \)

First, calculate \( (4x - 6)^2 \):

\[
(4x - 6)^2 = (4x)^2 - 2(4x)(6) + 6^2 = 16x^2 - 48x + 36
\]

Now substitute this into the equation for \( g(f(x)) \):

\[
g(f(x)) = 3(16x^2 - 48x + 36) - 3(4x - 6) - 1
\]

Simplify the terms:

\[
g(f(x)) = 48x^2 - 144x + 108 - 12x + 18 - 1
\]

Combine like terms:

\[
g(f(x)) = 48x^2 - 156x + 125
\]

Thus, \( (g \circ f)(x) = 48x^2 - 156x + 125 \).

#### Step 3: Find the inverse of \( (g \circ f)(x) \)

Now, to find the inverse, we need to solve for \( x \) in terms of \( y \), where \( y = 48x^2 - 156x + 125 \).

Start by setting \( y = 48x^2 - 156x + 125 \):

\[
y = 48x^2 - 156x + 125
\]

#### Step 4: Solve the quadratic equation

To solve for \( x \), we will use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the quadratic equation \( 48x^2 - 156x + (125 - y) = 0 \), the coefficients are:
- \( a = 48 \)
- \( b = -156 \)
- \( c = 125 - y \)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-156) \pm \sqrt{(-156)^2 - 4(48)(125 - y)}}{2(48)}
\]

Simplify the discriminant:

\[
x = \frac{156 \pm \sqrt{24336 - 19200 + 192y}}{96}
\]

\[
x = \frac{156 \pm \sqrt{4136 + 192y}}{96}
\]

Thus, the inverse function \((g \circ f)^{-1}(x)\) is:

\[
(g \circ f)^{-1}(x) = \frac{156 \pm \sqrt{4136 + 192x}}{96}
\]

#### Final Answer:

\[
(g \circ f)^{-1}(x) = \frac{156 \pm \sqrt{4136 + 192x}}{96}
\]

This is the inverse of the composition \( (g \circ f)(x) \).

---

Would you like further details or clarifications on any part of this solution?

Here are some related questions you could explore:
1. What is the domain of \( (g \circ f)(x) \)?
2. How do you verify if two functions are inverses of each other?
3. What is the graphical interpretation of inverse functions?
4. How do you find the inverse of a function without using the quadratic formula?
5. Can you find the inverse of the individual functions \( f(x) \) and \( g(x) \)?

**Tip**: When solving for the inverse of a function, remember to swap \( x \) and \( y \), and then solve for \( y \), which becomes your inverse function.

Fungsi f: R → R dan g: R → R dirumuskan dengan g(x) = 3x^2 - 3x - 1 dan f(x) = 4x - 6, maka (gof)^-1(x) = ...

Learn how to find the inverse of the composed function (g ∘ f)(x) with g(x) = 3x^2 - 3x - 1 and f(x) = 4x - 6. This step-by-step solution uses algebraic techniques including function composition and the quadratic formula. Suitable for high school students.

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