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

Hint: 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 .