Question

The function $f$ satisfies the functional equation $3f\left(x\right)+2f\left(\frac{x+59}{x-1}\right)=10x+30$ for all real $x\ne 1$. Then value of $f\left(7\right)$ is:

• A. $8$
• B. $4$
• C. $-8$
• D. $11$
• E. $44$

Answer 1

The correct option is B

$4$

Explanation for the correct option:

It is given that function $f$ satisfies the functional equation $3f\left(x\right)+2f\left(\frac{x+59}{x-1}\right)=10x+30$

Now, put $x=7$ in the given functional equation, we get

$3f\left(7\right)+2f\left(\frac{7+59}{7-1}\right)=10\left(7\right)+30$

$\Rightarrow 3f\left(7\right)+2f\left(11\right)=100$ ………….$\left(i\right)$

Now, put $x=11$ in the given functional equation, we get

$3f\left(11\right)+2f\left(\frac{11+59}{11-1}\right)=10\left(11\right)+30$

$\Rightarrow 3f\left(11\right)+2f\left(7\right)=140$ ………….$\left(ii\right)$

On solving $2\left(i\right)$ $-3\left(ii\right)$, we get

$\Rightarrow 4f\left(11\right)-9f\left(11\right)=200-420$

$\Rightarrow -5f\left(11\right)=-220$

$\Rightarrow f\left(11\right)=44$

Now, put $f\left(11\right)=44$ in equation $\left(i\right)$, we get,

$$\begin{gathered} \Rightarrow 3f\left(7\right)+2\left(44\right)=100 \\ \Rightarrow f\left(7\right)=\frac{12}{3} \end{gathered}$$

$\Rightarrow f\left(7\right)=4$

So, the value of $f\left(7\right)$ is $4$.

Hence, $\mathrm{Option}\left(\mathrm{B}\right)$ is the correct answer.