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# If $f(x)=\dfrac{3x+5}{2x-3}$, then $f\left( 15 \right)+f\left( -10 \right)=$ A. $\dfrac{1855}{629}$ B. $\dfrac{1875}{621}$ C. $\dfrac{1825}{621}$ D. $\dfrac{1875}{629}$

Last updated date: 13th Jun 2024
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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.

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.