Functions Test 42

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Functions Test 42

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Weekly Quiz Competition

Question 1
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The domain of the function $$f(x) = \dfrac{1} {\sqrt{\left \{ \sin {x} \right \} + \left \{ \sin (\pi + x) \right \}}}$$, where {.} denotes the fractional part, is

A
$$\left [0, \pi \right ]$$

B
$$\left (2n + 1 \right ) \pi/2, n \space \epsilon \space Z$$

C
$$\left (0, \pi \right )$$

D
None of these

Solution

d. $$f(x) = \dfrac{1} {\sqrt{\left \{ \sin {x} \right \} + \left \{ \sin (\pi + x) \right \}}} = \dfrac{1} {\sqrt{\left \{ \sin {x} \right \} + \left \{- \sin {x} \right \}}}$$

Now $$\left \{ \sin {x} \right \} + \left \{- \sin {x} \right \}$$ = $$\begin {cases}
0, & \sin x \space is \space an \space integer\\
1, & \sin x \space is \space not \space an \space integer
\end {cases}$$
For $$f(x)$$ to get defined $$\left \{ \sin {x} \right \} + \left \{- \sin {x} \right \} \neq 0$$
$$\Rightarrow \sin x \neq integer$$
$$\Rightarrow \sin x \neq \pm 1, 0$$
$$\Rightarrow x \neq \dfrac {n \pi} {2}, n \space \epsilon \space 1$$
Hence, the domain is $$R - \left \{\dfrac {n \pi} {2}\ \text{where}\ n \space \epsilon \space I \right \}$$
Question 2
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Domain (D) and range (R) of $$f(x) = \sin^{-1}\left (\cos^{-1} [x] \right )$$ where [.] denotes the greatest integer function is

A
$$D \equiv x \epsilon \left [ 1, 2 \right ), R \epsilon \left \{0 \right \}$$

B
$$D \equiv x \epsilon \left [ 0, 1 \right ], R \equiv \left \{-1, 0, 1 \right \}$$

C
$$D \equiv x \epsilon \left [ -1, 1 \right ), R \epsilon \left \{0, \sin^{-1} \left (\frac{\pi} {2}\right), \sin^{-1} (\pi) \right \}$$

D
$$D \equiv x \epsilon \left [-1, 1 \right ), R \epsilon \left \{-\frac{\pi} {2}, 0 , \frac{\pi} {2} \right \}$$

Solution
Question 3
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Directions For Questions

If $$a_{o} = x, a_{n+1} = f(a_n)$$, where n = 0, 1, 2,.....then answer the following question

...view full instructions

If f: $$R\rightarrow R$$ be given by $$f(x) = 3 + 4x$$ and $$a_n = A + Bx$$, then which of the following is not true?

A
A + B + 1 = $$2^{2n + 1}$$

B
| A - B| = 1`

C
$$\displaystyle \lim_{n \to \infty} \dfrac{A}{B} = -1$$

D
None of these

Solution

Since $$a_1 = g(x) = 3 + 4x$$
$$\therefore a_2 = g\{g^2(x)\} = g(3+4x) = 3 + 4(3+4x) = (4^2 - 1) + 4^x$$

$$a_3 = g\{g^(x)\} = g(15 + 4^2x) = 3 + 4 (15 + 4^2x) = 63 + 4^3x = (4^3 - 1) + 4^3 x$$

Similarly, we get $$a_n = (4^n - 1) + 4^n x$$
$$\Rightarrow A = 4^n - 1 \space and B = 4^n$$

$$\Rightarrow A + B + 1 = 2^{2n + 1}, |a - b| = 1\space and\space lim_{n \to \infty}\dfrac{4^n - 1}{4^n}$$

$$= lim_{n \to \infty}\left(1 - \dfrac{1}{4^n}\right) = 1$$
Question 4
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Directions For Questions

$$f(x) = \begin{cases} 2x + a, x \geq -1\\ bx^2 + 3, x < -1 \end{cases}$$

and $$g(x) = \begin{cases} x + 4, 0 \leq x \leq 4\\ -3x -2, -2 < x < 0 \end{cases}$$

...view full instructions

g(f(x)) is not defined if

A
$$a\space \epsilon ( 10, \infty) b\space \epsilon (5, \infty)$$

B
$$a\space \epsilon ( 4, 10) b\space \epsilon (5, \infty)$$

C
$$a\space \epsilon ( 10, \infty) b\space \epsilon (0, 1)$$

D
$$a\space \epsilon ( 4, 10) b\space \epsilon (1, 5)$$

Solution
Question 5
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The domain of $$f(x) = \sin^{-1} \left [ 2x^{2} - 3 \right ]$$, where [.]denotes the greatest integer function, is

A
$$\left ( - \frac{3} {2}, \frac{3} {2} \right )$$

B
$$\left ( - \frac{3} {2}, -1 \right ] \cup \left ( - \frac{5} {2}, \frac{5} {2} \right )$$

C
$$\left ( - \frac{5} {2}, \frac{5} {2} \right )$$

D
$$\left ( - \frac{5} {2}, -1 \right ] \cup \left [ 1, \frac{5} {2} \right )$$

Solution

We must have $$ -1 \leq \left [2x^{2} - 3 \right ] \leq 1$$

$$\Rightarrow -1 \leq 2x^{2} - 3 \leq 2 \Rightarrow 1 \leq x^{2} < \frac{5} {2}$$

$$\Rightarrow x \space \epsilon \left ( - \frac{5} {2}, -1 \right ] \cup \left [ 1, \frac{5} {2} \right )$$
Question 6
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Let $$f(x)$$ and $$g(x)$$ be differentiable for $$0\times < 1$$ such that $$f(0)=0, g(0), f(1)=6$$. Let there exist a real number $$c$$ in $$(0,1)$$ such that $$f'(c)=2g'(c)$$, then the value of $$g(1)$$ must be

A
$$1$$

B
$$3$$

C
$$2.5$$

D
$$-1$$

Solution
Question 7
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If g is the inverse of function $$f$$ and $$f'(x) = \frac{1}{1 + x}$$, then the value of g'(x) is equal to:

A
$$1 + x^7$$

B
$$\frac{1}{1 + [g(x)]^7}$$

C
$$1 + [g(x)]^7$$

D
$$7x^6$$

Solution

Since g is the inverse of $$f, f^{-1}(x) = g(x)$$
$$\therefore f[f^{-1}(x)] = f [g(x)] = x$$
$$\therefore f'[g(x)] \cdot \frac{d}{dx} [g(x)] = 1$$
$$\therefore f'[g(x)] \times g'(x) = 1$$
$$\therefore g'(x) = \frac{1}{f'[g(x)]}$$, where $$f'(x) = \frac{1}{1 + x^7}$$
$$\therefore g(x) = 1 + [g(x)]^7$$
Question 8
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Which one of the following is onto function define R to R .

A
$$ f(x) = |x| $$

B
$$ f(x) = e^{x}$$

C
$$ f(x) = x^{3}$$

D
$$ f(x) = \sin x $$

Solution
Question 9
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Which of the following in one -one function defined from R to R

A
$$ f(x) = |x| $$

B
$$ f(x) = \cos x $$

C
$$ f(x) =e^{x}$$

D
$$ f(x) = x^{2} $$
Question 10
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Which of the following is onto function-

A
$$ f : Z \rightarrow Z , f (x) = |x| $$

B
$$ f : N \rightarrow N , f (x) = |x| $$

C
$$ f : R_{0} \rightarrow R^{+} , f (x) = |x| $$

D
$$ f : C \rightarrow R , f (x) = |x| $$

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Re-Attempt Weekly Quiz Competition

Functions Test 42

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Functions Test 42

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SHARING IS CARING

Question 1

$$\left [0, \pi \right ]$$

$$\left (2n + 1 \right ) \pi/2, n \space \epsilon \space Z$$

$$\left (0, \pi \right )$$

None of these

Solution

Question 2

$$D \equiv x \epsilon \left [ 1, 2 \right ), R \epsilon \left \{0 \right \}$$

$$D \equiv x \epsilon \left [ 0, 1 \right ], R \equiv \left \{-1, 0, 1 \right \}$$

$$D \equiv x \epsilon \left [ -1, 1 \right ), R \epsilon \left \{0, \sin^{-1} \left (\frac{\pi} {2}\right), \sin^{-1} (\pi) \right \}$$

$$D \equiv x \epsilon \left [-1, 1 \right ), R \epsilon \left \{-\frac{\pi} {2}, 0 , \frac{\pi} {2} \right \}$$

Solution

Question 3

A + B + 1 = $$2^{2n + 1}$$

| A - B| = 1`

$$\displaystyle \lim_{n \to \infty} \dfrac{A}{B} = -1$$

None of these

Solution

Question 4

$$a\space \epsilon ( 10, \infty) b\space \epsilon (5, \infty)$$

$$a\space \epsilon ( 4, 10) b\space \epsilon (5, \infty)$$

$$a\space \epsilon ( 10, \infty) b\space \epsilon (0, 1)$$

$$a\space \epsilon ( 4, 10) b\space \epsilon (1, 5)$$

Solution

Question 5

$$\left ( - \frac{3} {2}, \frac{3} {2} \right )$$

$$\left ( - \frac{3} {2}, -1 \right ] \cup \left ( - \frac{5} {2}, \frac{5} {2} \right )$$

$$\left ( - \frac{5} {2}, \frac{5} {2} \right )$$

$$\left ( - \frac{5} {2}, -1 \right ] \cup \left [ 1, \frac{5} {2} \right )$$

Solution

Question 6

$$1$$

$$3$$

$$2.5$$

$$-1$$

Solution

Question 7

$$1 + x^7$$

$$\frac{1}{1 + [g(x)]^7}$$

$$1 + [g(x)]^7$$

$$7x^6$$

Solution

Question 8

$$ f(x) = |x| $$

$$ f(x) = e^{x}$$

$$ f(x) = x^{3}$$

$$ f(x) = \sin x $$

Solution

Question 9

$$ f(x) = |x| $$

$$ f(x) = \cos x $$

$$ f(x) =e^{x}$$

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

Question 10

$$ f : Z \rightarrow Z , f (x) = |x| $$

$$ f : N \rightarrow N , f (x) = |x| $$

$$ f : R_{0} \rightarrow R^{+} , f (x) = |x| $$

$$ f : C \rightarrow R , f (x) = |x| $$

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