To find \( g(x) \) such that \( g(f(x)) = 4x - 6 \), we start by substituting \( f(x) \) into the equation. Given that \( f(x) = 3x - 1 \), we can rewrite the equation as follows:
\[ g(f(x)) = g(3x - 1) = 4x - 6 \]
Now, we need to express \( x \) in terms of \( f(x) \). Let’s set \( y = f(x) = 3x - 1 \).
To solve for \( x \) in terms of \( y \), we can rearrange the equation:
\[ y = 3x - 1 \implies 3x = y + 1 \implies x = \frac{y + 1}{3} \]
Now, we substitute \( x = \frac{y + 1}{3} \) into \( 4x - 6 \):
\[ 4x - 6 = 4\left(\frac{y + 1}{3}\right) - 6 = \frac{4(y + 1)}{3} - 6 \]
Now simplify \( \frac{4(y + 1)}{3} - 6 \):
\[ \frac{4(y + 1)}{3} - 6 = \frac{4y + 4}{3} - \frac{18}{3} = \frac{4y + 4 - 18}{3} = \frac{4y - 14}{3} = \frac{4(y - \frac{14}{4})}{3} = \frac{4y - 14}{3} \]
Thus, we have:
\[ g(y) = \frac{4y - 14}{3} \]
Consequently, the function \( g(x) \) can be written as:
\[ g(x) = \frac{4x - 14}{3} \]
This is the required function \( g(x) \).
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