\( f\left(\frac{a x-1}{3}\right)=2 x+3 \) ve \( f\left(\frac{1}{3}\right)=7 \)

Let's dive into the world of function transformations with a twist! The function \( f \) gives us the value \( 7 \) when we plug in \( \frac{1}{3} \). This means that if we look at this transformation \( \frac{a x-1}{3} \), we can find a connection between \( a \) and \( x \) where the output lands us back to the original function value. To solve for \( a \), let's start with rearranging this expression. If we set \( \frac{a x - 1}{3} = \frac{1}{3} \), it follows that \( ax - 1 = 1 \) or \( ax = 2 \). So, when \( x = \frac{2}{a} \), we insert this back into our original function equation and verify if \( f\left(1\right)=2*\frac{2}{a}+3 \) leads us toward a consistent \( f \) value. Now that's some mathematical magic!