what if f(x) = √2x-1 and gof(x) = 3x⁴-x² what is g(x)

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?

Explore how to solve for g(x) from the composition of functions f(x) = √(2x - 1) and g∘f(x) = 3x⁴ - x². This step-by-step solution involves algebraic manipulation and function composition techniques, suitable for advanced high school students in Grades 11-12.

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