Functions Test 39

Result

Functions Test 39

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Weekly Quiz Competition

Question 1
1 / -0

Let $$f(x)=\dfrac{x}{(x^2-1)}$$. Then, dom $$(f)=?$$

A
$$R$$

B
$$R-\{1\}$$

C
$$R-\{-1\}$$

D
$$R-\{-1, 1\}$$

Solution

$$f(x)=\dfrac{x}{(x^2-1)}$$

$$f(x)$$ is not defined when $$(x^2-1)=0$$,

i.e., when $$(x-1)(x+1)=0$$,

i.e., when $$x=1$$ or $$x=-1$$.

$$\therefore dom(f)=R-[1, -1]$$.
Question 2
1 / -0

Let $$f(x)=\dfrac{\sin^{-1}x}{x}$$. Then, dom $$(f)=?$$

A
$$(-1, 1)$$

B
$$[-1, 1]$$

C
$$[-1, 1]-\{0\}$$

D
None of these

Solution
Question 3
1 / -0

If $$f(x)=\sqrt[3]{3-x^3}$$ then $$(f o f)(x)=?$$

A
$$x^{1/3}$$

B
$$x$$

C
$$(1-x^{1/3})$$

D
None of these

Solution
Question 4
1 / -0

Let $$f(x)=e^{\sqrt{x^2-1}}\cdot log (x-1)$$. Then, dom $$(f)=?$$

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

B
$$[-1, \infty)$$

C
$$(1, \infty)$$

D
$$(-\infty, -1]\cup (1, \infty)$$

Solution
Question 5
1 / -0

Let $$f(x)=\cos^{-1}2x$$. Then, dom $$(f)=?$$

A
$$[-1, 1]$$

B
$$\left[\dfrac{-1}{2}, \dfrac{1}{2}\right]$$

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

D
$$\left[\dfrac{-\pi}{4}, \dfrac{\pi}{4}\right]$$

Solution

$$\sin^{-1}2x$$ is defined only when $$2x\in[-1, 1]\Rightarrow x\in\left[\dfrac{-1}{2}, \dfrac{1}{2}\right]$$.
Question 6
1 / -0

Let $$ f : R \rightarrow R : f(x) =x +1 $$ and $$ g : R \rightarrow R : g(x) = 2x -3 $$.
Find $$(f +g) (x)$$.

A
$$3x -2$$

B
$$4x -5$$

C
$$3x -4$$

D
$$2x -3$$

Solution

Given ,
$$f(x)=x+1,g(x)=2x-3$$

$$\implies (f+g)x=f(x)+g(x)=x+1+2x-3=3x-2$$
Question 7
1 / -0

Let $$f(x)=\sqrt{\log(2x-x^2)}$$. Then, dom $$(f)=?$$

A
$$(0, 2)$$

B
$$[1, 2]$$

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

D
None of these

Solution

$$f(x)$$ is defined only when $$\log (2x-x^2) \geq 0$$

$$\Rightarrow (2x-x^2)\geq 1$$

$$\Rightarrow (1+x^2-2x)\leq 0$$

$$\Rightarrow (1-x)^2\leq 0$$

Since, any square no. can never be negative

$$\therefore (1-x)=0$$

$$\Rightarrow x=1$$.

$$\therefore dom(f)=\{1\}$$.
Question 8
1 / -0

Let $$f(x)=x^2$$. Then, dom $$(f)$$ and range(f) are respectively.

A
$$R$$ and $$R$$

B
$$R^+$$ and $$R^+$$

C
$$R$$ and $$R^+$$

D
$$R$$ and $$R-\{0\}$$

Solution
Question 9
1 / -0

If $$\displaystyle f(x) = | x - 2 |$$ and $$ g(x) = fof\,(x) $$ , then for $$ x > 20 , {g}\,'(x) = $$

A
$$ 2 $$

B
$$ 1 $$

C
$$ 3 $$

D
None of these

Solution
Question 10
1 / -0

If [.] denotes the greatest integer function then the domain of the real valued function $$\log_{[x+1/2]}\left| x^2 -x-2\right| $$ is

A
$$[\frac { 3 }{ 2 } ,\infty)$$

B
$$[\frac { 3 }{ 2 } ,2)\cup (2, \infty)$$

C
$$(\frac { 1 }{ 2 } ,2)\cup (2, \infty)$$

D
None of these

Solution