Functions Test 61

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Functions Test 61

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Question 1
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Let $$f(x)=\cos^{-1}(3x-1)$$. Then, dom $$(f)=?$$

A
$$\left(0, \dfrac{2}{3}\right)$$

B
$$\left[0, \dfrac{2}{3}\right]$$

C
$$\left[\dfrac{-2}{3}, \dfrac{2}{3}\right]$$

D
None of these

Solution

$$\cos^{-1}(3x-1)$$ is defined only when $$(3x-1)\in [-1, 1]$$
$$\Rightarrow 3x\in[0, 2]\Rightarrow x\in\left[0, \dfrac{2}{3}\right]\Rightarrow dom(f)=\left[0, \dfrac{2}{3}\right]$$.
Question 2
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Let $$f(x)=\sqrt{\cos x}$$. Then, dom $$(f)=?$$

A
$$\left[0, \dfrac{\pi}{2}\right]$$

B
$$\left[\dfrac{3\pi}{2}, 2\pi\right]$$

C
$$\left[0, \dfrac{\pi}{2}\right]\cup\left[\dfrac{3\pi}{2}, 2\pi\right]$$

D
None of these

Solution

$$f(x)$$ is defined only when $$\cos x \geq 0$$

$$\Rightarrow x$$ lies in $$1$$st or $$4$$th quadrant

$$\Rightarrow dom(f)=\left[0, \dfrac{\pi}{2}\right]\cup\left[\dfrac{3\pi}{2}, 2\pi\right]$$.
Question 3
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Let $$f(x)=log(1-x)+\sqrt{x^2-1}$$. Then, dom$$(f)=?$$

A
$$(1, \infty)$$

B
$$(-\infty, -1]$$

C
$$[-1, 1)$$

D
$$(0, 1)$$

Solution

Let $$f(x)=g(x)+h(x)$$, where $$g(x)=\log (1-x)$$ and $$h(x)=\sqrt{x^2-1}$$.

$$g(x)$$ is defined only when $$1-x > 0\Rightarrow x < 1$$. So, dom$$(g)=(-\infty, 1)$$.

$$h(x)$$ is defined only when $$x^2-1\geq 0\Rightarrow x\geq 1$$ or $$x\leq -1$$.

$$\therefore dom(h)=(-\infty, -1]\cup[1, \infty)$$.

$$\therefore dom(f)=dom(g)\cap dom(h)=(-\infty, -1]$$.
Question 4
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Directions For Questions

$$f(x) = \begin{cases} x-1, -1 \leq x \leq 0\\x^2, 0\leq x\leq 1 \end{cases}$$ and g(x) = sin x, consider the functions.
$$h_1(x) = f(|g(x)|) \space and \space h_2(x) = |f(g(x))|$$.

...view full instructions

Which of the following is not true about $$h_2(x)$$

A
Domain R

B
It periodic function with period $$2\pi$$

C
Range is [0, 1]

D
None of these

Solution
Question 5
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If $$ f(x)=\dfrac{2}{x-3}, g(x)=\dfrac{x-3}{x+4} $$ and $$ h(x)=-\dfrac{2(2 x+1)}{x^{2}+x-12} $$ then $$ \lim _{x \rightarrow 3}[f(x)+g(x)+h(x)] $$ is

A
$$ -2 $$

B
$$ -1 $$

C
$$ -\dfrac{2}{7} $$

D
0

Solution

c. We have $$ f(x)+g(x)+h(x)=\dfrac{x^{2}-4 x+17-4 x-2}{x^{2}+x-12} $$
$$ =\dfrac{x^{2}-8 x+15}{x^{2}+x-12}=\dfrac{(x-3)(x-5)}{(x-3)(x+4)} $$
$$ \therefore \lim _{x \rightarrow 3}[f(x)+g(x)+h(x)]=\lim _{x \rightarrow 3} \dfrac{(x-3)(x-5)}{(x-3)(x+4)}=-\dfrac{2}{7} $$.
Question 6
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For a real number y, let [y] denotes the greatest integer less than or equal to y. Then the function
$$ f(x)= \dfrac{tan(\pi \left[ x-\pi \right ])}{1+[x]^2} $$ is

A
discontinuous at some x

B
continuous at all x, but the derivative $$ f'(x_{0}) $$ does not exist for some x

C
$$ f'(x) $$ exist for all x but the derivative $$ f'(x_{0}) $$ does not exist second for some x

D
$$ f'(x) $$ exists for all x

Solution
Question 7
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Directions For Questions

Consider the function $$f(x) = f\left(\dfrac{x - 1}{x}\right) = 1 + x \space \forall \space x \space \epsilon \space R - {0, 1}\space and \space g(x) = 2f(x) - x + 1$$

...view full instructions

The number of roots of the equation g(x) = 1 is

A
2

B
1

C
3

D
0

Solution
Question 8
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The domain of the function $$f(x) = \sqrt{\ln_{(|x| - 1)} \left (x^{2} + 4x +4 \right )} $$ is

A
$$\left [ -3, -1 \right ] \cup \left [1, 2 \right ]$$

B
$$\left ( -3, -1 \right ) \cup \left [2, \infty \right )$$

C
$$\left ( -\infty, -3 \right ] \cup \left (-2, -1 \right ) \cup \left (2, \infty \right )$$

D
None of these

Solution
Question 9
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If $$f : X \rightarrow Y$$, where $$X$$ and $$Y$$ are sets containing natural numbers, $$f(x) = \frac{x + 5} {x + 2}$$ then the number of elements in the domain and range of $$f(x) are respectively

A
1 and 1

B
2 and 1

C
2 and 2

D
1 and 2

Solution
Question 10
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The domain of definition of the function $$f(x)$$ given by the equation $$2^x + 2^y = 2$$ is (IIT-JEE, 2000)

A
$$0 < x \leq 1$$

B
$$0 \leq x \leq 1$$

C
$$-\infty < x \leq 0$$

D
$$-\infty < x < 1$$

Solution