Domain and Range of a Function

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JEE Main 2012

LEVELJEE Main

The equation $e^{s i n x} - e^{- s i n x} - 4 = 0$ has

(A)

infinite number of real roots

(B)

no real roots

(C)

exactly one real root

(D)

exactly four real roots

JEE Main 2011

LEVELJEE Main

The domain of the function $f (x) = \frac{1}{∣ x ∣ - x}$ is

(A)

(0, \infty)

(B)

(- \infty, 0)

(C)

(- \infty, \infty) ∖ {0}

(D)

(- \infty, \infty)

JEE Main 2007

LEVELJEE Main

The largest interval lying in $(- π /2, π /2)$ for which the function, $f (x) = 4^{- x^{2}} + cos^{- 1} (x /2 - 1) + lo g (cos x)$ , is defined, is

(A)

[- π /4, π /2]

(B)

[0, π /2]

(C)

[0, π]

(D)

(- π /2, π /2)

JEE Main 2004

LEVELJEE Main

The range of the function $f (x) =^{7 - x} P_{x - 3}$ is

(A)

{1, 2, 3, 4, 5}

(B)

{1, 2, 3, 4, 5, 6}

(C)

{1, 2, 3, 4}

(D)

{1, 2, 3}

JEE Main 2004

LEVELJEE Main

If $f : R \to S$ , defined by $f (x) = sin x - 3 cos x + 1$ , is onto, then the interval of $S$ is

(A)

[- 1, 3]

(B)

[- 1, 1]

(C)

[0, 1]

(D)

[0, 3]

JEE Main 2004

LEVELJEE Main

The domain of the function $f (x) = \frac{s i n ^{- 1} ( x - 3 )}{9 - x ^{2}}$ is

(A)

[1, 2]

(B)

[2, 3)

(C)

[1, 2]

(D)

[2, 3]

JEE Main 2003

LEVELJEE Main

Domain of definition of the function $f (x) = \frac{3}{4 - x ^{2}} + lo g_{10} (x^{3} - x)$ , is

(A)

(- 1, 0) \cup (1, 2) \cup (2, \infty)

(B)

(a, 2)

(C)

(- 1, 0) \cup (a, 2)

(D)

(1, 2) \cup (2, \infty)

JEE Advanced 2005

LEVELJEE Main

$X$ and $Y$ are two sets and $f : X \to Y$ . If ${f (c) = y; c \subset X, y \subset Y}$ and ${f^{- 1} (d) = x; d \subset Y, x \subset X}$ , then the true statement is

(A)

f (f^{- 1} (b)) = b

(B)

f^{- 1} (f (a)) = a

(C)

f^{- 1} (f (b)) = b, b \subset Y

(D)

f^{- 1} (f (a)) = a, a \subset X

JEE Advanced 2003

LEVELBoard

Range of the function $f (x) = \frac{x ^{2} + x + 2}{x ^{2} + x + 1}, x \in R$ is

(A)

(1, \infty)

(B)

(1, 11/7]

(C)

(1, 7/3]

(D)

[1, 7/5]

JEE Advanced 2001

LEVELJEE Main

The domain of definition of $f (x) = \frac{l o g _{2} ( x + 3 )}{x ^{2} + 3 x + 2}$ is

(A)

R ∖ {- 1, - 2}

(B)

(- 2, \infty)

(C)

R ∖ {- 1, - 2, - 3}

(D)

(- 3, \infty) ∖ {- 1, - 2}

JEE Advanced 2000

LEVELJEE Main

The domain of definition of the function given by the equation $2^{x} + 2^{y} = 2$ is

(A)

0 < x \leq 1

(B)

0 \leq x \leq 1

(C)

- \infty < x \leq 0

(D)

- \infty < x < 1

JEE Advanced 1990

LEVELJEE Main

The number of solutions of the equation $sin (e^{x}) = 5^{x} + 5^{- x}$ is

The domain of definition of the function $y = \frac{1}{l o g _{10} ( 1 - x )} + x + 2$ is

(A)

(- 3, - 2)

excluding

- 2.5

(B)

[0, 1]

excluding

0.5

(C)

[- 2, 1)

excluding

0

A function $f : I R \to I R$ , where $I R$ is the set of real numbers, is defined by $f (x) = \frac{α x ^{2} + 6 x - 8}{α + 6 x - 8 x ^{2}}$ . Find the interval of values of $α$ for which $f$ is onto. Is the function one-to-one for $α = 3$ ? Justify your answer.

JEE Advanced 1985

LEVELJEE Main

If $f (x) = sin ln (\frac{4 - x ^{2}}{1 - x})$ , then domain of $f (x)$ is .... and its range is .........

JEE Advanced 1983

LEVELJEE Main

The larger of $cos (ln θ)$ and $ln (cos θ)$ if $- π /2 < θ < π /2$ is .........

JEE Advanced 1983

LEVELJEE Main

The values of $f (x) = 3 sin \frac{π ^{2}}{16} - x^{2}$ lie in the interval .........

JEE Advanced 1978

LEVELJEE Main

Find the domain and range of the function $f (x) = \frac{x ^{2}}{1 + x ^{2}}$ . Is the function one-to-one?

JEE Advanced 1982

LEVELJEE Main

Show that the equation $e^{s i n x} - e^{- s i n x} - 4 = 0$ has no real solution.

JEE Advanced 2007

LEVELJEE Main

Let $f (x) = \frac{x ^{2} - 6 x + 5}{x ^{2} - 5 x + 6}$ . Match of expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

- 1 < x < 1

, then

f (x)

satisfies

(Q)

1 < x < 2

, then

f (x)

satisfies

(R)

3 < x < 5

, then

f (x)

satisfies

(S)

x > 5

, then

f (x)

satisfies

List-II

(1)

0 < f (x) < 1

(2)

f (x) < 0

(3)

f (x) > 0

(4)

f (x) < 1

JEE Main 2019 (11 January)

LEVELJEE Main

Let $f : R \to R$ be defined by $f (x) = \frac{x}{1 + x ^{2}}, x \in R$ . Then the range of $f$ is :

(A)

(- 1, 1) - {0}

(B)

[\frac{- 1}{2}, \frac{1}{2}]

(C)

R - [\frac{- 1}{2}, \frac{1}{2}]

(D)

R - [- 1, 1]

JEE Main 2019 (09 April Shift 1)

LEVELJEE Main

If the function $f : R - {1, - 1} \to A$ defined by $f (x) = \frac{x ^{2}}{1 - x ^{2}}$ , is surjective, then $A$ is equal to

(A)

R - [- 1, 0)

(B)

R - (- 1, 0)

(C)

R - {- 1}

(D)

[0, \infty)

JEE Main 2019 (9 April)

LEVELJEE Main

The domain of the definition of the function $f (x) = \frac{1}{4 - x ^{2}} + lo g_{10} (x^{3} - x)$ is :-

(A)

(1, 2) \cup (2, \infty)

(B)

(- 1, 0) \cup (1, 2) \cup (3, \infty)

(C)

(- 1, 0) \cup (1, 2) \cup (2, \infty)

(D)

(- 2, - 1) \cup (- 1, 0) \cup (2, \infty)

JEE Main 2019 (10 April Shift 1)

LEVELJEE Main

Let $f (x) = x^{2}, x \in R$ . For any $A \subseteq R$ , define $g (A) = {x \in R, f (x) \in A}$ . If $S = [0, 4]$ , then which one of the following statements is not true ?

(A)

f (g (S)) \neq = f (S)

(B)

f (g (S)) = S

(C)

g (f (S)) = g (S)

(D)

g (f (S)) \neq = S

JEE Main 2020 (8 January Shift 2)

LEVELJEE Main

Let $f : (1, 3) \to R$ be a function defined by $f (x) = \frac{x [ x ]}{x ^{2} + 1}$ , where $[x]$ denotes the greatest integer $\leq x$ . Then the range of $f$ is:

(A)

(\frac{2}{5}, \frac{3}{5}] \cup (\frac{3}{4}, \frac{4}{5})

(B)

(\frac{2}{5}, \frac{4}{5}]

(C)

(\frac{3}{5}, \frac{4}{5})

(D)

(\frac{2}{5}, \frac{1}{2}) \cup (\frac{3}{5}, \frac{4}{5}]

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The largest interval lying in (−π/2,π/2) for which the function, f(x)=4−x2+cos−1(x/2−1)+log(cosx), is defined, is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f:R→S, defined by f(x)=sinx−3​cosx+1, is onto, then the interval of S is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The domain of the function f(x)=9−x2​sin−1(x−3)​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Domain of definition of the function f(x)=4−x23​+log10​(x3−x), is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X and Y are two sets and f:X→Y. If {f(c)=y;c⊂X,y⊂Y} and {f−1(d)=x;d⊂Y,x⊂X}, then the true statement is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Range of the function f(x)=x2+x+1x2+x+2​,x∈R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The domain of definition of f(x)=x2+3x+2log2​(x+3)​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The domain of definition of the function given by the equation 2x+2y=2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the equation sin(ex)=5x+5−x is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The domain of definition of the function y=log10​(1−x)1​+x+2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=sinln(1−x4−x2​​), then domain of f(x) is .... and its range is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The larger of cos(lnθ) and ln(cosθ) if −π/2<θ<π/2 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The values of f(x)=3sin16π2​−x2​ lie in the interval .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the domain and range of the function f(x)=1+x2x2​. Is the function one-to-one?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Show that the equation esinx−e−sinx−4=0 has no real solution.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be defined by f(x)=1+x2x​,x∈R. Then the range of f is :

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The domain of the definition of the function f(x)=4−x21​+log10​(x3−x) is :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=x2,x∈R. For any A⊆R, define g(A)={x∈R,f(x)∈A}. If S=[0,4], then which one of the following statements is not true ?

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The equation $e^{s i n x} - e^{- s i n x} - 4 = 0$ has

The domain of the function $f (x) = \frac{1}{∣ x ∣ - x}$ is

The largest interval lying in $(- π /2, π /2)$ for which the function, $f (x) = 4^{- x^{2}} + cos^{- 1} (x /2 - 1) + lo g (cos x)$ , is defined, is

The range of the function $f (x) =^{7 - x} P_{x - 3}$ is

If $f : R \to S$ , defined by $f (x) = sin x - 3 cos x + 1$ , is onto, then the interval of $S$ is

The domain of the function $f (x) = \frac{s i n ^{- 1} ( x - 3 )}{9 - x ^{2}}$ is

Domain of definition of the function $f (x) = \frac{3}{4 - x ^{2}} + lo g_{10} (x^{3} - x)$ , is

$X$ and $Y$ are two sets and $f : X \to Y$ . If ${f (c) = y; c \subset X, y \subset Y}$ and ${f^{- 1} (d) = x; d \subset Y, x \subset X}$ , then the true statement is

Range of the function $f (x) = \frac{x ^{2} + x + 2}{x ^{2} + x + 1}, x \in R$ is

The domain of definition of $f (x) = \frac{l o g _{2} ( x + 3 )}{x ^{2} + 3 x + 2}$ is

The domain of definition of the function given by the equation $2^{x} + 2^{y} = 2$ is

The number of solutions of the equation $sin (e^{x}) = 5^{x} + 5^{- x}$ is

The domain of definition of the function $y = \frac{1}{l o g _{10} ( 1 - x )} + x + 2$ is

If $f (x) = sin ln (\frac{4 - x ^{2}}{1 - x})$ , then domain of $f (x)$ is .... and its range is .........

The larger of $cos (ln θ)$ and $ln (cos θ)$ if $- π /2 < θ < π /2$ is .........

The values of $f (x) = 3 sin \frac{π ^{2}}{16} - x^{2}$ lie in the interval .........

Find the domain and range of the function $f (x) = \frac{x ^{2}}{1 + x ^{2}}$ . Is the function one-to-one?

Show that the equation $e^{s i n x} - e^{- s i n x} - 4 = 0$ has no real solution.

Let $f : R \to R$ be defined by $f (x) = \frac{x}{1 + x ^{2}}, x \in R$ . Then the range of $f$ is :

If the function $f : R - {1, - 1} \to A$ defined by $f (x) = \frac{x ^{2}}{1 - x ^{2}}$ , is surjective, then $A$ is equal to

The domain of the definition of the function $f (x) = \frac{1}{4 - x ^{2}} + lo g_{10} (x^{3} - x)$ is :-

Let $f (x) = x^{2}, x \in R$ . For any $A \subseteq R$ , define $g (A) = {x \in R, f (x) \in A}$ . If $S = [0, 4]$ , then which one of the following statements is not true ?

Let $f : (1, 3) \to R$ be a function defined by $f (x) = \frac{x [ x ]}{x ^{2} + 1}$ , where $[x]$ denotes the greatest integer $\leq x$ . Then the range of $f$ is: