Question

The function $f$ satisfies the functional equation $3f\left( x \right) + 2f\left[ {\dfrac{{x + 59}}{{x - 1}}} \right] = 10x + 30$ for all real $x \ne 1$ . Then value of $f\left( 7 \right)$ is
$\left( 1 \right)$ $8$
$\left( 2 \right)$ $4$
$\left( 3 \right)$ $- 8$
$\left( 4 \right)$ $11$
$\left( 5 \right)$ $44$

Answer 1

We have to find the value of the function $f$ at $x = 7$ . We solve this question using the concept of solving the linear equations . We should have the knowledge of the concept of elimination method for solving the given functional expression . First we will find the relation for the given expression at $x = 7$ . On putting the value $x = 7$ we will obtain an expression in terms of another value of $x$ . Then we will put the value of $x$ as the value , which we will obtain from the functional equation at $x = 7$ . And then we will substitute the values of the two functional expressions such that we obtain the value for the functional expression at $x = 7$ .

Complete answer: Given :
$3f\left( x \right) + 2f\left[ {\dfrac{{x + 59}}{{x - 1}}} \right] = 10x + 30$ for all real $x \ne 1$

We have to find the value of $f\left( 7 \right)$ .

Now , we will be putting the value of $x$ as $7$ in the given functional expression .

On putting $x = 7$ , we get the functional expression as :
$3f\left( 7 \right) + 2f\left[ {\dfrac{{7 + 59}}{{7 - 1}}} \right] = 10 \times 7 + 30$

On solving the functional equation , we get the expression as :
$3f\left( 7 \right) + 2f\left[ {\dfrac{{66}}{6}} \right] = 70 + 30$

Further , we get
$3f\left( 7 \right) + 2f\left[ {11} \right] = 100 - - - \left( 1 \right)$

Now , as we got the other value of $x$ as $11$ in the simplified function expression , we will put the value of $x$ as $11$ for the other relation of the functional equation .

Putting the value of x as 11 in the given functional expression , we get the value as :
$3f\left( {11} \right) + 2f\left[ {\dfrac{{11 + 59}}{{11 - 1}}} \right] = 10 \times 11 + 30$
On solving the functional equation , we get the expression as :
$3f\left( {11} \right) + 2f\left[ {\dfrac{{70}}{{10}}} \right] = 110 + 30$
Further , we get
$3f\left( {11} \right) + 2f\left[ 7 \right] = 140 - - - \left( 2 \right)$
Now , we will solve the two equations for the value of $f\left( 7 \right)$ using the elimination method .
Multiplying equation $\left( 1 \right)$ by $3$ , we get the expression as :
$3 \times \left[ {3f\left( 7 \right) + 2f\left[ {11} \right] = 100} \right]$
$9f\left( 7 \right) + 6f\left[ {11} \right] = 300 - - - \left( 3 \right)$
Multiplying equation $\left( 2 \right)$ by $2$ , we get the expression as :
$2 \times \left[ {3f\left( {11} \right) + 2f\left[ 7 \right] = 140} \right]$
$6f\left( {11} \right) + 4f\left[ 7 \right] = 280 - - - \left( 4 \right)$
Subtracting equation $\left( 4 \right)$ from equation $\left( 3 \right)$ , we get the value of the functional expression as :
$9f\left( 7 \right) + 6f\left[ {11} \right] - \left( {6f\left( {11} \right) + 4f\left[ 7 \right]} \right) = 300 - 280$
On solving , we get
$5f\left( 7 \right) = 20$
$f\left( 7 \right) = 4$
Hence , we get the value of the functional expression at $x = 7$ as $4$ .
Thus , the correct option is $\left( 2 \right)$ .

Note:
For the value of the functional expression , we can use any of the methods of solving the equation . We could also use the substitution method or the cross multiplication method to solve the value of the functional expression . But we used the elimination method and it is easier and less complicated than the other two methods .