2025 Nov TZ1 P2 Q7

Consider the function \(f(x) = \frac{3x + 1}{2x - 4}\) where \(x \neq 2\).

1. Write down the equation of

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   1. the vertical asymptote of the graph of \(f\) ;

   2. the horizontal asymptote of the graph of \(f\).

2. Find \(f'(x)\).

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3. Hence, explain briefly why \(f\) is never increasing.

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The point \(\mathrm{P} (x, f(x) )\) is on the graph of \(f\) such that \(x < 2\).
Let \(d\) be the distance between the points \(\mathrm{P}\) and \( \mathrm{Q} (3, 5)\).

4. Find an expression for \(d\) in terms of \(x\).

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5. Hence, find the coordinates of \(\mathrm{P}\) such that \(d\) is a minimum.

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