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**Hint**: We have given a function $ f(x)=\dfrac{3x+5}{2x-3} $ and we have to find the value of $ f\left( 15 \right)+f\left( -10 \right)= $ we put the value of $ x=15 $ and $ x=-10 $ in the given function one by one and calculate the value of $ f\left( 15 \right) $ and $ f\left( -10 \right) $ . Then, add both to obtain a desired result.

**:**

__Complete step-by-step answer__We have given that $ f(x)=\dfrac{3x+5}{2x-3} $

We have to find the value of $ f\left( 15 \right)+f\left( -10 \right) $ .

Now, first we calculate the value of $ f(15) $ , for this we put the value $ x=15 $ in the given equation.

When we put value, we get

$ \begin{align}

& f(x)=\dfrac{3x+5}{2x-3} \\

& f(15)=\dfrac{3\times 15+5}{2\times 15-3} \\

\end{align} $

Now, simplify the equation to solve further

\[\begin{align}

& f(15)=\dfrac{45+5}{30-3} \\

& f(15)=\dfrac{50}{27}.......(i) \\

\end{align}\]

Similarly, we calculate the value of $ f(-10) $ .

When we put the value $ x=-10 $ in the given equation, we get

$ \begin{align}

& f(x)=\dfrac{3x+5}{2x-3} \\

& f(-10)=\dfrac{3\times \left( -10 \right)+5}{2\times \left( -10 \right)-3} \\

& f(-10)=\dfrac{-30+5}{-20-3} \\

& f(-10)=\dfrac{-25}{-23} \\

& f(-10)=\dfrac{25}{23}.................(ii) \\

\end{align} $

Now, to find the value of $ f\left( 15 \right)+f\left( -10 \right) $ , substitute the value from equation (i) and equation (ii)

$ f\left( 15 \right)+f\left( -10 \right)=\dfrac{50}{27}+\dfrac{25}{23} $

Now, take LCM to solve further, as $ 27 $ and $ 23 $ don’t have common factor, we directly cross multiply to solve

$ \begin{align}

& f\left( 15 \right)+f\left( -10 \right)=\dfrac{50\times 23+27\times 25}{27\times 23} \\

& f\left( 15 \right)+f\left( -10 \right)=\dfrac{1150+675}{621} \\

& f\left( 15 \right)+f\left( -10 \right)=\dfrac{1825}{621} \\

\end{align} $

Hence, the value of $ f\left( 15 \right)+f\left( -10 \right)=\dfrac{1825}{621} $.

**So, the correct answer is “Option C”.**

**Note**: One may relate this question with the differentiation as the given equation is of the form $ f(x)=\dfrac{3x+5}{2x-3} $. But we have to find the value of $ f\left( 15 \right)+f\left( -10 \right) $, which are not derivatives so it is not a question of differentiation. If it is asked to find the value of $ f'\left( 15 \right)+f'\left( -10 \right) $ instead of $ f\left( 15 \right)+f\left( -10 \right) $, then we first differentiate the given equation and find the values.

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