A function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \frac{2x+3}{3x+5}$.

1. Express the target form as a single fraction to compare with the original function.

$a + \frac{b}{3x+5} = \frac{a(3x+5) + b}{3x+5} = \frac{3ax + 5a + b}{3x+5}$

2. Equate the numerators and compare the coefficients of $x$ and the constant terms.

Since $\frac{2x+3}{3x+5} = \frac{3ax + 5a + b}{3x+5}$, we equate the numerators:
$2x + 3 = 3ax + (5a + b)$

Comparing the coefficients of $x$:
$3a = 2 \implies a = \frac{2}{3}$

Comparing the constant terms:
$5a + b = 3$

3. Solve for $b$ and state the final expression.

Substitute $a = \frac{2}{3}$ into the second equation:
$5\left(\frac{2}{3}\right) + b = 3$
$\frac{10}{3} + b = 3$
$b = 3 - \frac{10}{3} = -\frac{1}{3}$

Therefore, $a = \frac{2}{3}$ and $b = -\frac{1}{3}$, giving:
$\mathrm{f}(x) = \frac{2}{3} + \frac{-1/3}{3x+5}$

1. Determine the constants $a$ and $b$ from part (i) to express the function in a form suitable for reading transformations.

$\mathrm{f}(x) = \frac{2x+3}{3x+5}$
Using algebraic manipulation to separate the numerator:
$2x+3 = \frac{2}{3}(3x+5) - \frac{10}{3} + 3 = \frac{2}{3}(3x+5) - \frac{1}{3}$
$\mathrm{f}(x) = \frac{\frac{2}{3}(3x+5) - \frac{1}{3}}{3x+5} = \frac{2}{3} - \frac{1/3}{3x+5}$
So, $a = \frac{2}{3}$ and $b = -\frac{1}{3}$.

2. Determine the sequence of transformations affecting the $x$-coordinates.

To transform the base graph $y = \frac{1}{x}$ into $y = \frac{1}{3x+5}$:

1. Translate the graph by $5$ units in the negative $x$-direction (replacing $x$ with $x+5$). This gives $y = \frac{1}{x+5}$.
2. Scale the graph parallel to the $x$-axis by a factor of $\frac{1}{3}$ (replacing $x$ with $3x$). This gives $y = \frac{1}{3x+5}$.

3. Determine the sequence of transformations affecting the $y$-coordinates.

To transform $y = \frac{1}{3x+5}$ into $y = \frac{2}{3} - \frac{1/3}{3x+5}$:
3. Scale the graph parallel to the $y$-axis by a factor of $-\frac{1}{3}$ (which is equivalent to scaling by $\frac{1}{3}$ and reflecting in the $x$-axis). This gives $y = -\frac{1/3}{3x+5}$.
4. Translate the graph by $\frac{2}{3}$ units in the positive $y$-direction. This gives $y = \frac{2}{3} - \frac{1/3}{3x+5} = \mathrm{f}(x)$.