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 \[\forall x\in R\], 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:
\[\Rightarrow f\left( x+2 \right)=2f\left( x \right)-f\left( x+1 \right)\] - (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.