Functions - Maharashtra Board STD 12 Maths Questions

Q 1MCQ1 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.

Q 2MCQ1 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 3MCQ1 Mark

If $f: R \rightarrow R$ is given by $f(x)=x^3+3$, then $f^{-1}(x)$ is equal to:

A
$\text{x}^\frac{1}{3}-3$
B
$\text{x}^\frac{1}{3}+3$
✓
$(\text{x}-3)^\frac{1}{3}$
D
$\text{x}+3^\frac{1}{3}$

Answer: C.

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Q 4MCQ1 Mark

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

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

Answer: C.

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Q 5MCQ1 Mark

If $g(x)=x^2+x-2$ and $\frac{1}{2} g \circ f(x)=2 x^2-5 x+2$, then $f(x)$ is equal to:

✓
$2x - 3$
B
$2x + 3$
C
$2x^2 + 3x + 1$
D
$2x^2 - 3x - 1$

Answer: A.

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Q 6Solve the Following Question.(2 Marks)2 Marks

Let $\text{f(x)=}\begin{cases}1+\text{x,}&0\leq\text{x}\leq2\\3-\text{x,}&2<\text{x}\leq3\end{cases}.$ Find fof.

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Q 7Solve the Following Question.(2 Marks)2 Marks

If $f : R \rightarrow R$ is defined by $f(x) = x^2$, find $f^{-1}(-25)$.

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Q 8Solve the Following Question.(2 Marks)2 Marks

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.

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Q 9Solve the Following Question.(2 Marks)2 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 10Solve the Following Question.(2 Marks)2 Marks

If $f(x) = 2x + 5$ and $g(x) = x^2 + 1$ be two real functions, then describe the following functions:
fog
Also, show that fof $\neq f^2$

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Q 11Solve the Following Question.(3 Marks)3 Marks

Show that f : R → R, given by f(x) = x - [x], is neither one-one nor onto.

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Q 12Solve the Following Question.(3 Marks)3 Marks

State with reasons whether the following functions have inverse:
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

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Q 13Solve the Following Question.(3 Marks)3 Marks

Let A = [-1, 1]. Then, discuss whether the following functions from A to itself are one-one, onto or bijective:
g(x) = |x|

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Q 14Solve the Following Question.(3 Marks)3 Marks

Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R,$ defined by $f(x) = x^3 + 1$

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Q 15Solve the Following Question.(3 Marks)3 Marks

Classify the following functions as injection, surjection or bijection:
f : R → R, defined by f(x) = |x|

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Q 16Solve the Following Question.(5 Marks)5 Marks

Let $\text{f}:[-1,\infty)\rightarrow[-1,\infty)$ be given by $f(x) = (x + 1)^2 - 1, \text{x}\geq-1.$ Show that f is invertible. Also, find the set $S = \{x : f(x) = f^{-1}(x)\}.$

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Q 17Solve the Following Question.(5 Marks)5 Marks

Consider $\text{f}:\text{R}_+\rightarrow[-5,\infty)$ given by $f(x) = 9x^2 + 6x - 5.$ Show that f is invertible with $\text{f}^{-1}(\text{x})=\frac{\sqrt{\text{x}+6}-1}{3}.$

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Q 18Solve the Following Question.(5 Marks)5 Marks

If $f : Q \rightarrow Q, g : Q \rightarrow Q$ are two functions defined by $f(x) = 2x$ and $g(x) = x + 2$, show that f and g are bijective maps. Verify that $(gof)^{-1} = f^{-1}og^{-1}$.

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Q 19Solve the Following Question.(5 Marks)5 Marks

If $\text{f(x)}=\sqrt{1-\text{x}}$ and $\text{g(x)}=\log_\text{e}\text{x}$ are two real functions, then describe, functions fog and gof.

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Q 20Solve the Following Question.(5 Marks)5 Marks

Let $f : N \rightarrow N$ be a function as$ f(x) = 9x^2 + 6x - 5$. Show that $f : N \rightarrow S$, where S is the range of f, is invertible. Find the inverse of f and hence find $f^{-1}(43$) and $f^{-1}(163).$

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