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.