General Tips for Choosing Function

Q. A function f (x) satisfies the following property:f (xy)= flx) f(y). Show that the function f (x) is continuous forall values of x if it is continuous at x 1.

Q. 3. A real valued function f(x) satisfies the equation f(x+y) = f(x) f(a-y) + f(y) f(a-x), where x, y belongs to R , where a is a constant and f(0) = 0 , then f(2a-x) is (1) -f(x) (2) f(x) (3) f(a-x) (4) f(-x)

Q.

Let f(x) = 2x + 1. Then the number of real values of x for which the three numbers f(x), f(2x), f(4x) are in G.P. is

Q. 16. A Function F(x) satisfies the functional equation x F(x) + F(1-x) = 2x - x, for all real x. What should be F(x).

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 $f\left(1003\right)=\frac{1}{K}$, then $K$ equals

1. $1003$
2. $2003$
3. $2005$
4. $2006$

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 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(xy) = f(x)+f(y), and f(e) = 1, then find the value of $f\left(e^2\right)$

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Q. For real numbers x and y, the relation f satisfies f(xy) = f(x).f(y) and f(0)≠0. Find the value of f(2019).

Q.

If f(xy) = f(x)+f(y), and f(e) = 1, then find the value of $f\left(e^2\right)$

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Q. let A={-2, -1, 0, 1, 0} and f:A->Z be a function defined by f(x)=x² - 2x - 3. find a. range of f and b.pre-images of 6, -3, 5

Q. If $f\left(x_1\right)-f\left(x_2\right)=f\left(\frac{x_1-x_2}{1-x_1{x}_2}\right)$ for $x_1,x_2ϵ$ (-1, 1), then f(x) is

1. $log\frac{1-x}{1+x}$
2. $log\frac{2+x}{1-x}$
3. $ta{n}^{-1}\frac{1-x}{1+x}$
4. $ta{n}^{-1}\frac{1+x}{1-x}$

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

1. $1$
2. $2$
3. more than $2$ but finite
4. infinitely many

Q. A real valued function f(x) satisfies the functional equation 4f(x) + 5f(6 – x) = x2 + 5. Then the value of 9f(2) is equal to

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

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

Q. A function f(x) satisfies f(x + y) = f(x) + y for all value x, y belongs to real number and f(0)=5. then f(2020) is equal to

Q. A real valued function f(x) satisfies the functional equation f(x-y)=f(x)f(y)-f(a-x).f(a+y) where a is given constant and f(0)=1 , f(2a-x)= (1) -f(x) (2) f(x) (3) f(a-x) (4) f(-x)

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

The graph of $y=g\left(x\right)$ in its domain is broken at

1. $1$ point
2. $2$ points
3. $3$ points
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(x) = x², find $\frac{f\left(1.1\right)-f\left(1\right)}{\left(1.1\right)-1}$.

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. For every pair of number x and y the function f satisfies 2f(x-y)= f(x)f(y) then the value of f(2) isa.2b.1/2c.0d.4

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

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

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 f(x+y) = f(x) . f(y) , then

1. $f\left(x\right)=a^{kx}$

2. $f\left(x\right)=0$

3. $f\left(x\right)=kx$

4. $f\left(x\right)=x^n$