Functions - Gujarat Board (GSEB) STD 11 Science MATHS Questions

Q 1MCQ1 Mark

The domain of definition of the function $\text{f(x)}=\sqrt{\text{x}-1}+\sqrt{3-\text{x}}$ is:

A
$[1,\infty)$
B
$\big(-\infty,3\big)$
C
$(1,3)$
✓
$\big[1,3\big]$

Answer: D.

Q 2MCQ1 Mark

If $\text{x}\neq1$ and $\text{f(x)}=\frac{\text{x}+1}{\text{x}-1}$ is a real function, then $\text{f}(\text{f}(\text{f(2)}))$ is:

A
1
B
2
✓
3
D
4

Answer: C.

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Q 3MCQ1 Mark

The domain of definition of $\text{f(x)}=\sqrt{\frac{\text{x}+3}{(2-\text{x})(\text{x}-5)}}$ is:

✓
$(-\infty,-3]\cup(2,5)$
B
$(-\infty,-3]\cup(2,5)$
C
$(-\infty,-3]\cup[2,5]$
D
None of these.

Answer: A.

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Q 4MCQ1 Mark

The range of the function $\text{f(x)}=\frac{\text{x}^2-\text{x}}{\text{x}^2+2\text{x}}$ is:

A
$\text{R}$
B
$\text{R}-\{1\}$
✓
$\text{R}-\Big\{\frac{1}{2},1\Big\}$
D
None of these.

Answer: C.

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Q 5MCQ1 Mark

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=2 x+3$ and $g(x)=x^2+7$, then the values of $x$ such that $g(f(x))=8$ are:

A
1, 2
B
-1, 2
✓
-1, -2
D
1, -2

Answer: C.

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Q 6(Each question 1 marks)1 Mark

Let $\text{f(x)}=\frac{\alpha\text{x}}{\text{x}+1},\text{x}\neq-1.$ Then write the value of $\alpha$ satisfying $\text{f}(\text{f(x)})=\text{x}$ for all $\text{x}\neq-1$

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Q 7(Each question 1 marks)1 Mark

If f is a real function satisfying $\text{f}\Big(\text{x}+\frac{1}{\text{x}}\Big)=\text{x}^2+\frac{1}{\text{x}^2}$ for all $\text{x}\in\text{R}-\{0\},$ then write the expression for f(x).

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Q 8(Each question 1 marks)1 Mark

Write the domain and range of $\text{f(x)}=\sqrt{\text{x}-[\text{x}]}$

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Q 9(Each question 1 marks)1 Mark

Let A and B be two sets such that n(A) = p and n(B) = q, write the number of functions from A to B.

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Q 10(Each question 1 marks)1 Mark

What is the fundamental difference between a relation and a function? Is every relation a function?

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Q 11(Each question 2 marks)2 Marks

Let A = {12, 13, 14, 15, 16, 17} and f : A → Z be a function given by f(x) = highest prime factor of x. Find range of f.

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Q 12(Each question 2 marks)2 Marks

If $\text{f(x)}=\frac{2\text{x}}{1+\text{x}^2},$ show that $\text{f}(\tan\theta)=\sin2\theta$

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Q 13(Each question 2 marks)2 Marks

If $f: R \rightarrow R$ be defined by $f(x)=x^2+1$, then find $f^{-1}\{17\}$ and $f^{-1}\{-3\}$.

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Q 14(Each question 2 marks)2 Marks

Let A = {9, 10, 11, 12, 13} and let f : A → N be defined by f(n) = the highest prime factor of n. Find the range of f.

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Q 15(Each question 2 marks)2 Marks

Find the domain and range of the following real valued functions: $\text{f(x)}=\frac{\text{ax}+\text{b}}{\text{bx}-\text{a}}$

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Q 16(Each question 3 marks)3 Marks

If $\text{f(x)}=\frac{\text{x}-1}{\text{x}+1},$ then show that:

$\text{f}\Big(\frac{1}{\text{x}}\Big)=-\text{f(x)}$
$\text{f}\Big(-\frac{1}{\text{x}}\Big)=-\frac{1}{\text{f(x)}}$

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Q 17(Each question 3 marks)3 Marks

If $\text{f(x)}=\begin{cases}\text{x}^2,&\text{when }\text{ x}<0\\\text{x},&\text{when }\ 0\leq\text{x}<1\\\frac{1}{\text{x}},&\text{when }\text{ x}>0\end{cases}$ Find:

$\text{f}\Big(\frac{1}{2}\Big)$
$\text{f}(-2)$
$\text{f}(1)$
$\text{f}(\sqrt{3})$
$\text{f}(\sqrt{-3})$

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Q 18(Each question 3 marks)3 Marks

If $\text{y}=\text{f(x)}=\frac{\text{ax}-\text{b}}{\text{bx}-\text{a}},$ show that x = f(y).

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Q 19(Each question 3 marks)3 Marks

If $\text{f(x)}=\log_\text{e}(1-\text{x})$ and $\text{g(x)}=[\text{x}],$ then determine the following functions: (fg)(0)

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Q 20(Each question 3 marks)3 Marks

Find $\text{f}+\text{g},\text{ f}-\text{g},\text{ cf}(\text{c}\in\text{ R},\text{c}\neq0),\text{ fg},\frac{1}{\text{f}}$ and $\frac{\text{f}}{\text{g}}$ in the following: If $f(x) = x^3 + 1$ and $g(x) = x + 1$

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Q 21(Each question 4 marks)4 Marks

If for non-zerox, $\text{af(x)}+\text{bf}\Big(\frac{1}{\text{x}}\Big)=\frac{1}{\text{x}}-5,$ where $\text{a}\neq\text{b},$ then find f(x).

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Q 22(Each question 4 marks)4 Marks

Let $\text{f}:[0,\infty)\rightarrow\text{R}$ and $\text{g}:\text{R}\rightarrow\text{R}$ be defined by $\text{f(x)}=\sqrt{\text{x}}$ and g(x) = x. Find f + g, g - g, fg and $\frac{\text{f}}{\text{g}}$

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Q 23(Each question 4 marks)4 Marks

Let f and g be two real functions defined by $\text{f(x)}=\sqrt{\text{x}+1}$ and $\text{g(x)}=\sqrt{9-\text{x}^2}$ Then describe the following functions: $\frac{\text{g}}{\text{f}}$

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Q 24(Each question 4 marks)4 Marks

Let f and g be two real functions defined by $\text{f(x)}=\sqrt{\text{x}+1}$ and $\text{g(x)}=\sqrt{9-\text{x}^2}$ Then describe the following functions: $2\text{f}-\sqrt{5}\text{g}$

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Q 25(Each question 4 marks)4 Marks

If $\text{f(x)}=\log_\text{e}(1-\text{x})$ and $\text{g(x)}=[\text{x}],$ then determine the following functions: $\frac{\text{f}}{8}$

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Functions MATHS - Functions

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