Given that:

• $f(x) = \sqrt{2x - 1}$
• $g \circ f(x) = 3x^4 - x^2$

We are asked to find $g(x)$.

The expression $g \circ f(x)$ represents the composition of functions, so:

$g(f(x)) = 3x^4 - x^2$

Step 1: Replace $f(x)$ with its expression in $g(f(x))$

Since $f(x) = \sqrt{2x - 1}$, we can rewrite the composition as:

$g(\sqrt{2x - 1}) = 3x^4 - x^2$

Step 2: Let $y = f(x) = \sqrt{2x - 1}$

Now, express the equation in terms of $y$:

$g(y) = 3x^4 - x^2$

Step 3: Solve for $x$ in terms of $y$

From $y = \sqrt{2x - 1}$, square both sides to eliminate the square root:

$y^2 = 2x - 1$

Solving for $x$:

$x = \frac{y^2 + 1}{2}$

Step 4: Substitute this into $g(y)$

Now substitute $x = \frac{y^2 + 1}{2}$ into the expression for $g(y)$:

$g(y) = 3 \left( \frac{y^2 + 1}{2} \right)^4 - \left( \frac{y^2 + 1}{2} \right)^2$

Step 5: Simplify the expression for $g(y)$

You can expand both terms and simplify, but the expression might become quite complex. However, the general form of $g(y)$ is now established in terms of $y$.

Would you like to go further with the simplification, or do you have any questions so far?