Question

The function $$f$$ satisfies the functional equation $$3f\left(x\right)+2f\left[{\displaystyle \frac{x+59}{x-1}}\right]=10x+30$$ 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 :
$$3f\left(x\right)+2f\left[{\displaystyle \frac{x+59}{x-1}}\right]=10x+30$$ 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 :
$$3f\left(7\right)+2f\left[{\displaystyle \frac{7+59}{7-1}}\right]=10\times 7+30$$

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

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 :
$$3f\left(11\right)+2f\left[{\displaystyle \frac{11+59}{11-1}}\right]=10\times 11+30$$
On solving the functional equation , we get the expression as :
$$3f\left(11\right)+2f\left[{\displaystyle \frac{70}{10}}\right]=110+30$$
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 .