General Tips for Choosing Function

Q. If $f\left(x\right)$ is a polynomial function satisfying
$f\left(x\right)f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ and $f\left(1\right)=2$ then the number of such functions possible is/are

Q. If a function $f:Z\to Z$ is defined as $mf\left(x\right)+f\left(my\right)=f\left(f\right(x+y\left)\right)$, where $m$ is a non-zero constant, then which of the following is/are correct?

1. $f\left(x\right)=mx+n,n\in Z$
2. $f\left(x\right)=nx+m,n\in Z$
3. $f\left(x\right)=0$
4. $f\left(1\right),f\left(4\right),f\left(7\right),f\left(10\right)$ are in A.P.

Q. If $f\left(x\right)$ is a polynomial of degree $n$ such that $f\left(0\right)=0,f\left(1\right)=\frac{1}{2},....,f\left(n\right)=\frac{n}{n+1}$, then the value of $f(n+1)$ is

1. $1$ when $n$ is odd
2. $\frac{n}{n+2}$ when $n$ is even
3. $-\frac{n}{n+1}$ when $n$ is odd
4. $-1$ when $n$ is even

Q.

If $f\left(x\right)$ be a polynomial function satisfying $f\left(x\right)·f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ and $f\left(4\right)=65$. Then find $f\left(6\right)$.

1. $65$

2. $217$

3. $63$

4. none of these

Q. A real-valued function f(x) satisfies the equation $f(x-y)=f\left(x\right)f\left(y\right)-f(a-x)f(a+y)$ where a is a constant and f(0) = 1. Then f(2a - x) is equal to

1. -f(x)
2. f(x)
3. f(a)+f(a-x)=0
4. f(-x)

Q. If $F(n+1)=\frac{2F\left(n\right)+1}{2}$ for $n=1,2,\dots$ and $F\left(1\right)=2$, then $F\left(101\right)$ equals

Q. Let $f\left(x\right)=f_1\left(x\right)-2f_2\left(x\right)$,

where, $f_1\left(x\right)=\{\begin{array}{cc}min\{x^2,|x|\},& |x|\le 1\\ max\{x^2,|x|\},& |x|>1\end{array}$

and, $f_2\left(x\right)=\{\begin{array}{cc}min\{x^2,|x|\},& |x|>1\\ max\{x^2,|x|\},& |x|\le 1\end{array}$

and, $g\left(x\right)=\{\begin{array}{c}min\left\{f\right(t):-3\le t\le x,-3\le x<0\}\\ max\left\{f\right(t):0\le t\le x,0\le x\le 3\}\end{array}$

For $x\in (-1,0)$, $f\left(x\right)+g\left(x\right)$ is

1. $x^2-2x+1$
2. $x^2-2x-1$
3. $x^2+2x-1$
4. $x^2+2x+1$

Q.

If f(x) satisfies f(x+y) = f(x) + f(y) and f(5) = 15 then find f(3)?

1. 15

2. 9

3. 6

4. 21

Q. If $F(n+1)=\frac{2F\left(n\right)+1}{2}$ for $n=1,2,\dots$ and $F\left(1\right)=2$, then $F\left(101\right)$ equals

Q. Let f be a function satisfying the functional equation $f\left(x\right)+2f\left(\frac{2x+1}{x-2}\right)=3x,x\ne 2$; then, the value of f(7) is

1. 5
2. -5
3. 1
4. -1

Q.

If $f:R\to R$ satisfies $f(x+y)=f\left(x\right)+f\left(y\right)$, for all $x,y\in R$ and $f\left(1\right)=7$ , then ${\sum }_{r=1}^{n}f\left(r\right)$ is

1. $\frac{7n(n+1)}{2}$

2. $\frac{7n}{2}$
   

3. $\frac{7(n+1)}{2}$
   

4. 7n+(n+1)

Q. If a function $f:Z\to Z$ is defined as $mf\left(x\right)+f\left(my\right)=f\left(f\right(x+y\left)\right)$, where $m$ is a non-zero constant, then which of the following is/are correct?

1. $f\left(x\right)=mx+n,n\in Z$
2. $f\left(x\right)=nx+m,n\in Z$
3. $f\left(x\right)=0$
4. $f\left(1\right),f\left(4\right),f\left(7\right),f\left(10\right)$ are in A.P.

Q. Let $f:N\to R$ be a function satisfying the following conditions:
$f\left(1\right)=1$ and $f\left(1\right)+2f\left(2\right)+\dots +nf\left(n\right)=n(n+1)f\left(n\right)$ for $n\ge 2$.
If $f\left(999\right)=\frac{1}{K}$, then $K$ equals

1. $999$
2. $1000$
3. $1998$
4. $2000$

Q. If the function $f$ satisfies the relation $f(x+y)+f(x-y)=2f\left(x\right)f\left(y\right)\forall x,y\in R$ and $f\left(0\right)\ne 0$, then

1. $f\left(x\right)$ is an even function
2. $f\left(x\right)$ is an odd function
3. If $f\left(2\right)=a$, then $f(-2)=a$
4. If $f\left(4\right)=b$, then $f(-4)=-b$

Q. If a function F is such that F(0)= 2, F (1) = 3, F (x + 2) = 2 F(x) – F (x -1) for x $\le$ 0, then F(5) is equal to

1. -7
2. -3
3. 17
4. 13

Q. If $f\left(x\right)$ is a polynomial of degree $n$ such that $f\left(0\right)=0,f\left(1\right)=\frac{1}{2},....,f\left(n\right)=\frac{n}{n+1}$, then the value of $f(n+1)$ is

1. $1$ when $n$ is odd
2. $\frac{n}{n+2}$ when $n$ is even
3. $-\frac{n}{n+1}$ when $n$ is odd
4. $-1$ when $n$ is even