Classification of Functions

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f

R

f

R

f

R

f

R

- f (x)

f (x)

f (a) + f (a - x)

f (- x)

f (x) = - f (- x)

f (2 + x) = f (2 - x)

f (x) = f (- x)

f (x + 2) = f (x - 2)

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

\frac{7 n}{2}

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

7 n + (n + 1)

∣ sin 3 x ∣ + sin^{2} x

cos x + cos^{2} x

cos 4 x + tan^{2} x

cos 2 x + sin x

f (x)

f (x)

f (x)

f (x)

x

1

f (x)

g (x)

14

16

12

8

f (x)

f (1/ x) \to 0

x \to 0

x f (x) \to 1

x \to 0

f (x) = ln x

f (π /2) = - 1

f (π) = 1

f (- π) = 0

f (π /4) = 1

g (x) = \pm 1 - x^{2}

g (x) = 1 - x^{2}

g (x) = - 1 - x^{2}

g (x) = 1 + x^{2}

f (x^{2}) = (f (x))^{2}

f (x + y) = f (x) + f (y)

f (∣ x ∣) = ∣ f (x) ∣

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)+2f(x1​)=3x,x=0 and S={x∈R:f(x)=f(−x)}; then S:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For real x, let f(x)=x3+5x+1, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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 a given constant and f(0)=1,f(2a−x) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The graph of the function y=f(x) is symmetrical about the line x=2, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f:R→R satisfies f(x+y)=f(x)+f(y), for all x,y∈R and f(1)=7, then ∑r=1n​f(r) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A function f from the set of natural numbers to integers defined by f(n)={2n−1​,−2n​,​when n is oddwhen n is even​ is

Which one is not periodic

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:(−π/2,π/2)→R be given by f(x)=(log(secx+tanx))3. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f:[0,3]→[1,29], defined by f(x)=2x3−15x2+36x+1, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the functions f(x) and g(x) are defined on R→R such that f(x)={0,x,​x∈rationalx∈irrational​; g(x)={0,x,​x∈irrationalx∈rational​ then (f−g)(x) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f:[0,∞)→[0,∞), and f(x)=1+xx​ then f is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let function f:R→R be defined by f(x)=2x+sinx for x∈R, then f is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let g(x)=1+x−[x] and f(x)=⎩⎨⎧​−1,0,1,​x<0x=0x>0​. Then for all x, f(g(x)) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let E={1,2,3,4} and F={1,2}. Then the number of onto functions from E to F is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x) be defined for all x>0 and be continuous. Let f(x) satisfy f(x/y)=f(x)−f(y) for all x,y and f(e)=1. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=cos[π2]x+cos[−π2]x, where [x] stands for the greatest integer function, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let g(x) be a function defined on [−1,1]. If the area of the equilateral triangle with two of its vertices at (0,0) and [x,g(x)] is 43​​, then the function g(x) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f and g be increasing and decreasing functions, respectively from [0,∞) to [0,∞). Let h(x)=f(g(x)). If h(0)=0, then h(x)−h(1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=∣x−1∣. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R be the set of real numbers. If f:R→R is a function defined by f(x)=x2, then f is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let {x} and [x] denotes the fractional and integral part of a real number x respectively. Solve 4{x}=x+[x].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } There are exactly two distinct linear functions, ........., and ......... which map [−1,1] onto [0,2].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A be a set of n distinct elements. Then the total number of distinct functions from A to A is ......... and out of these ......... are onto functions.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A discontinuous function y=f(x) satisfying x2+y2=4 is given by f(x)=.........

If $f (x) + 2 f (\frac{1}{x}) = 3 x, x \neq = 0$ and $S = {x \in R : f (x) = f (- x)}$ ; then S:

For real $x$ , let $f (x) = x^{3} + 5 x + 1$ , then

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 a given constant and $f (0) = 1, f (2 a - x)$ is equal to

The graph of the function $y = f (x)$ is symmetrical about the line $x = 2$ , then

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

A function $f$ from the set of natural numbers to integers defined by $f (n) = {\frac{n - 1}{2}, - \frac{n}{2}, when n is odd when n is even$ is

Let $f : (- π /2, π /2) \to R$ be given by $f (x) = (lo g (sec x + tan x))^{3}$ . Then

The function $f : [0, 3] \to [1, 29]$ , defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 1$ , is

If the functions $f (x)$ and $g (x)$ are defined on $R \to R$ such that $f (x) = {0, x, x \in rational x \in irrational$ ; $g (x) = {0, x, x \in irrational x \in rational$ then $(f - g) (x)$ is

If $f : [0, \infty) \to [0, \infty)$ , and $f (x) = \frac{x}{1 + x}$ then $f$ is

Let function $f : R \to R$ be defined by $f (x) = 2 x + sin x$ for $x \in R$ , then $f$ is

Let $g (x) = 1 + x - [x]$ and $f (x) = ⎩ ⎨ ⎧ - 1, 0, 1, x < 0 x = 0 x > 0$ . Then for all $x$ , $f (g (x))$ is equal to

Let $E = {1, 2, 3, 4}$ and $F = {1, 2}$ . Then the number of onto functions from $E$ to $F$ is

Let $f (x)$ be defined for all $x > 0$ and be continuous. Let $f (x)$ satisfy $f (x / y) = f (x) - f (y)$ for all $x, y$ and $f (e) = 1$ . Then

If $f (x) = cos [π^{2}] x + cos [- π^{2}] x$ , where $[x]$ stands for the greatest integer function, then

Let $g (x)$ be a function defined on $[- 1, 1]$ . If the area of the equilateral triangle with two of its vertices at $(0, 0)$ and $[x, g (x)]$ is $\frac{3}{4}$ , then the function $g (x)$ is

Let $f$ and $g$ be increasing and decreasing functions, respectively from $[0, \infty)$ to $[0, \infty)$ . Let $h (x) = f (g (x))$ . If $h (0) = 0$ , then $h (x) - h (1)$ is

Let $f (x) = ∣ x - 1∣$ . Then

Let $R$ be the set of real numbers. If $f : R \to R$ is a function defined by $f (x) = x^{2}$ , then $f$ is :

Let ${x}$ and $[x]$ denotes the fractional and integral part of a real number $x$ respectively. Solve $4 {x} = x + [x]$ .

There are exactly two distinct linear functions, ........., and ......... which map $[- 1, 1]$ onto $[0, 2]$ .

Let $A$ be a set of $n$ distinct elements. Then the total number of distinct functions from $A$ to $A$ is ......... and out of these ......... are onto functions.

A discontinuous function $y = f (x)$ satisfying $x^{2} + y^{2} = 4$ is given by $f (x) = .........$