Definition of Functions

Q. If $f\left(x\right)=\sqrt{4-x^2}+\frac{1}{\sqrt{|sinx|-sinx}}$, then the domain of f(x) is

1. [–2, 0]
2. (0, 2]
3. [–2, 2}
4. [–2, 0)

Q. The value of the parameter $\alpha$ , for which the function $f\left(x\right)=1+\alpha x,\alpha \ne 0$ is the inverse of itself, is

1. -2
2. 1
3. -1
4. 2

Q.

The relation f is defined by

$f\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 2\\ 3x,& 3\le x\le 10\end{array}$

The relation g is defined by

$g\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 3\\ 3x,& 2\le x\le 10\end{array}$

Show that f is a function and g is not a function.

Q. Let f be the subset of Z × Z defined by f = {( ab , a + b ): a , b ∈ Z }. Is f a function from Z to Z : justify your answer.

Q. Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true? (i) f is a relation from A to B (ii) f is a function from A to B. Justify your answer in each case.

Q.

Let $f:R\to R:f\left(x\right)=x^2+3$. Find the pre-images of each of the following under f :

(i) 19

(ii) 28

(iii) 2

Q.

Find the value of $e^{i2\pi }$ .

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Q.

Draw the graph of logarithm function y= ${\log }_{a}x$ When x>0and a>1.

1. 

2. 

3. 

4. 

Q.

Is There a Function That Is Both Even And Odd?

1. True
2. False

Q.

If $y=f\left(x\right),$ where $f:A\to B$ is a function in $x,$ then which of the following statements is true?

(i) Every element of $A$ needs to have an image.

(ii) $x\in A$ must be related to one and only one value of $y$ of $B.$

1. Statement (i) is true, but statement (ii) is false.

2. Statement (ii) is true, but statement (i) is false.

3. Neither statement (i) nor statement (ii) is true.

4. Both statements (i) and (ii) are true.

Q.

If $r=\alpha b\times c+\beta c\times a+\gamma a\times b$ and $\left[abc\right]=2$, then $\alpha +\beta +\gamma$ is equal to

1. $r·[b\times c+c\times a+a\times b]$

2. $\frac{1}{2}r·(a+b+c)$

3. $2r·(a+b+c)$

4. $4$

Q.

How many of the following functions are even [sin x is odd and cosx is even]

(a) f(x) = $x^2|x|$ (b) f(x) = $e^x+e^{-x}$

(c) f(x) = log[$\frac{1-x}{1+x}$] (d) log($\sqrt{x^2+1}$- x)

(e) f(x) = log(x + $\sqrt{x^2+1}$ (f) $a^x-a^{-x}$

(g) f(x) = $\sin x+\cos x$ (h) $\sin x\times (e^x-e^{-x})$

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Q. If $f$ is a function such that $f\left(0\right)=2,$ $f\left(1\right)=3$ and $f(x+2)=2f\left(x\right)-f(x+1)$ for every real $x,$ then the value of $f\left(5\right)$ is

1. $7$
2. $13$
3. $1$
4. $5$

Q.

Show that the statement

p : "If x is a real number such that $x^3+4x=0$ then x is 0" is true by (i) direct method (ii) Method of contrapositive.

Q. Among the following which is odd function

1. 
2. 
3. 
4. 

Q. 22. A function f:R->R satisfies the following conditions: 1)f(x) not equal to zero , for all values of x corresponding to R 2)f(x+y) = f(x).f(y) , for all values of x & y corresponding to R 3)f(x) is differentiable 4)f(0)=2 The value of f(0) is - A)1 B)-1 C)2 D)1/2

Q. If $f\left(x\right)=4x-x^2,x\in R$, then $f(a+1)-f(a-1)$ is equal to

1. $2(4-a)$
2. $4(2-a)$
3. $4(2+a)$
4. $2(4+a)$

Q.

The integral part of $(6\sqrt{6}+14{)}^{201}$ is even

1. True

2. False

Q.

A parabola has its axis of symmetry as the x axis. The parabola opens to the left, so its equation is of the type

1. y² =4ax , a>0

2. x²=4ay , a>0

3. y² = - 4ax , a>0

4. x²= -4ay , a>0

Q. Let $A$ and $B$ are two matrices of same order $3\times 3$ given by $A=\left[\begin{array}{ccc}1& 3& \lambda +2\\ 2& 4& 6\\ 3& 5& 8\end{array}\right]$ and $B=\left[\begin{array}{ccc}3& 2& 4\\ 3& 2& 5\\ 2& 1& 4\end{array}\right].$
If $Tr(AB{)}^T+Tr(BA{)}^T=Tr\left(BA\right),$ then the value of $2\lambda$ equals to

1. $-103$
2. $203$
3. $-203$
4. $103$

Q.

Which of the following statements is/are true about an odd function f(x)?

1. f(-x) = f(x)

2. f(-x) = - f(x)

3. Its graph is symmetric about the origin

4. Its graph is symmetric about the x-axis

5. Its graph is symmetric about the y-axis

Q. If f(x+2y, x-2y)=xy, then f(x, y) equals

1. 
2. 
3. 
4. 

Q. Let $A=\{a,b,c,d,e\}$ and $B=\{1,2,3,4,5\}$. The number of relations which are not functions from $A$ to $B$ is

1. $2^{25}-5^5$
2. $2^{20}-5^5$
3. $5^5-25$
4. $2^{25}-5\times 3$

Q.

How many of the following functions are even [sin x is odd and cosx is even]

(a) f(x) = $x^2|x|$ (b) f(x) = $e^x+e^{-x}$

(c) f(x) = log[$\frac{1-x}{1+x}$] (d) log($\sqrt{x^2+1}$- x)

(e) f(x) = log(x + $\sqrt{x^2+1}$ (f) $a^x-a^{-x}$

(g) f(x) = $\sin x+\cos x$ (h) $\sin x\times (e^x-e^{-x})$

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Q. Pick the graph(s) which corresponds to a function.

1. 
2. 
3. 
4. 

Q.

$\text{If}y=si{n}^{-1}x,then(1-x^2)y_2-x{y}_1$

1. 2

2. 2y

3. 1

4. 0

Q.

If $g\left\{f\right(x\left)\right\}=|sinx|$ and $f\left\{g\right(x\left)\right\}=(sin\sqrt{x}{)}^2$, then

1. $f\left(x\right)=sinx,g\left(x\right)=|x|$

2. $f\left(x\right)=si{n}^{2}x,g\left(x\right)=\sqrt{x}$

3. $f\left(x\right)=x^2,g\left(x\right)=sin\sqrt{x}$

4. f and g cannot be determined

Q.

In $\Delta$ ABC, if $\frac{1}{b+c}+\frac{1}{c+a}=\frac{3}{a+b+c},$ then C is equal to

1. 90°

2. 45°

3. 60°

4. 30°

Q. what is linear equation

Q.

The common thing about these three graphs is that, they are all _______ in their domain.

1. Continuous

2. Differentiable except at one point

3. Differentiable

4. Continuous except at one point