Functions - Rajasthan Board (RBSE) STD 12 Science MATHS Questions

Q 1M.C.Q (1 Marks)1 Mark

Let $\text{A}=\{\text{x}\in\text{R}:-1\leq\text{x}\leq1\}=\text{B}.$ Then, the mapping $f : A \rightarrow B$ given by $f(x) = x|x|$ is:

A
Injective but not surjective.
B
Surjective but not injective.
✓
Bijective.
D
None of these.

Answer: C.

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Q 2M.C.Q (1 Marks)1 Mark

The range of the function $\text{f(x)}=^{7-\text{x}}\text{P}_{\text{x}-3}$ is:

A
$\{1, 2, 3, 4, 5\}$
B
$\{1, 2, 3, 4, 5, 6\}$
C
$\{1, 2, 3, 4\}$
✓
$\{1, 2, 3\}$

Answer: D.

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Q 3M.C.Q (1 Marks)1 Mark

Let $\text{f(x)}=\frac{1}{1-\text{x}}.$ Then, {fo(fof)}(x):

x for all $\text{x}\in\text{R}$
x for all $\text{x}\in\text{R}-\{1\}$
x for all $\text{x}\in\text{R}-\{0,1\}$
None of these.

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Q 4M.C.Q (1 Marks)1 Mark

If $\text{g(f(x))}=|\sin\text{x}|$ and $\text{f(g(x))}=(\sin\sqrt{\text{x}})^2,$ then

$\text{f(x)}=\sin^2\text{x},\ \text{g(x)}=\sqrt{\text{x}}$
$\text{f(x)}=\sin\text{x},\ \text{g(x)}=|\text{x}|$
$\text{f(x)}=\text{x}^2,\ \text{g(x)}=\sin\sqrt{\text{x}}$
$\text{f and g cannot be determined.}$

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Q 5M.C.Q (1 Marks)1 Mark

The function $\text{f}:[0,\infty)\rightarrow\ \text{R}$ given by $\text{f(x)}=\frac{\text{x}}{\text{x}+1}$ is:

One-one and onto.
One-one but not onto.
Onto but not one-one.
Onto but not one-one.

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Q 62 Marks Questions2 Marks

Let $\text{f}:\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\rightarrow\ \text{A}$ be defined by f(x)= sinx. If f is a bijection, write set A.

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Q 72 Marks Questions2 Marks

Let $\text{f}:\Big(-\frac{\pi}{2},\frac{\pi}{2}\Big)\rightarrow\ \text{R}$ be a function defined by f(x) = cos[x]. write range (f).

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Q 82 Marks Questions2 Marks

What is the range of the function $\text{f(x)}=\frac{|\text{x}-1|}{\text{x}-1}?$

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Q 92 Marks Questions2 Marks

If $A = \{1, 2, 3, 4\}$ and $B = \{a, b, c, d\}$ define any four bijections from $A$ to $B.$ Also give their inverse functions.

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Q 102 Marks Questions2 Marks

If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| – x, $\forall\ \text{x}\in\text{R}.$ Then find fog and gof. Hence find fog(-3), fog(5) and gof(-2).

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Q 113 Marks Question3 Marks

Find $fog$ and $gof$ if : $f(x) = e^x, g(x) = \log_ex$

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Q 123 Marks Question3 Marks

If $\text{f(x)}=\sqrt{\text{x}+3}$ and $g(x) = x^2 + 1$ be two real functions, then find $fog$ and $gof$.

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Q 133 Marks Question3 Marks

Find fog and gof if:f(x) = x + 1, g(x) = sinx

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Q 143 Marks Question3 Marks

Find fog and gof if:$\text{f}(\text{x})=\text{c},\text{c}\in \text{R},\text{g(x)}=\sin \text{x}^2$

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Q 153 Marks Question3 Marks

If $f : R \rightarrow (0, 2)$ defined by $\text{f(x)}=\frac{\text{e}^{\text{x}}-\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}+1$ is invertible, find $f^{-1}.$

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Q 165 Marks Questions5 Marks

The relation S defined on the set R of all real number by the rule aSb iff a ≥ b is: