Question

If f (x) is a function such that f (0) = 2, f (1) = 3 and f (x + 2) = 2 f (x) – f (x + 1) for $$\mathrm{\forall }x\in R$$, then f (5) is:
(A). 5
(B). 13
(C). 3
(D). -5

Answer 1

Hint: First write all the known relations and values at once. Now substitute x = 0 in the relation. Use the above known values, given in the question to find the unknown. Now increase the value of x by 1 repeat the process. Repeat all the steps till you get the functional value which you require f (5). Stop at this point which is the required result.

Complete step-by-step solution -
The first functional value given in the question in the question is:
$$\Rightarrow f\left(0\right)=2$$ - (1)
The second functional value given in the question is:
$$\Rightarrow f\left(1\right)=3$$ - (2)
The relation between f and x given in the question is:
$$\Rightarrow f(x+2)=2f\left(x\right)-f(x+1)$$ - (3)
By substituting x = 0 in above equation, we get it as:
$$\Rightarrow f\left(2\right)=2f\left(0\right)-f\left(1\right)$$
By substituting the functional values, we get it as follows:
$$\Rightarrow f\left(2\right)=2\left(2\right)-3$$
By simplifying the above equation, we get it as follows:
$$\Rightarrow f\left(2\right)=4-3=1$$
By substituting x = 1 in equation (3), we get it as follows:
$$\Rightarrow f\left(3\right)=2f\left(1\right)-f\left(2\right)$$
By substituting the function values in above equation, we get it as:
$$\Rightarrow f\left(3\right)=2\left(3\right)-1$$
By simplifying the above equation, we get it as:
$$\Rightarrow f\left(3\right)=6-1=5$$
By substituting x = 2 in equation (3), we get the equation as:
$$\Rightarrow f\left(4\right)=2f\left(3\right)-f\left(2\right)$$
By substituting the functional values in above equation, we get:
$$\Rightarrow f\left(4\right)=2\left(5\right)-1$$
By simplifying the above equation, we get it as follows:
$$\Rightarrow f\left(4\right)=10-1=9$$
By substituting x = 3 in equation (3), we get the equation as:
$$\Rightarrow f\left(5\right)=2f\left(4\right)-f\left(3\right)$$
By simplifying the functional values, we get it as follows:
$$\Rightarrow f\left(5\right)=2\left(9\right)-5$$
By simplifying the above equation, we get it as follows:
$$\Rightarrow f\left(5\right)=18-5=13$$
So, the value of f (5) is 13 such that it satisfies the given condition.
Therefore option (b) is the correct answer for the given question.

Note: Be careful while calculating functional value: every functional value is dependent on others indirectly. So, if you make mistakes in any functional value, the whole answer might be wrong. Generally, by seeing the f (0), f (1) values students have a misconception of f (x) = x + 2 and they solve it, but it is a very big mistake. Even if you assume like this you must first verify in the relation do not blindly solve by the assumption.