Definition of functions

Q.

$\text{The function}$ $f:R+\to (1,e)\text{defined by}f\left(x\right)=\frac{X^2+e}{X^2+1}$ is

1. Both one-one and onto
2. Neither one-one nor onto
3. One-one but not onto
4. Onto but not one-one

Q.

The function $f\left(x\right)=2x^3-3x^2+90x+174$ is increasing in the interval.

1. $\frac{1}{2}<x<1$

2. $\frac{1}{2}<x<2$

3. $3<x<\frac{59}{4}$

4. $-\infty <x<\infty$

Q.

For what values of $x,$ function $f\left(x\right)=x^4-4x^3+4x^2+40$is monotonic decreasing?

1. $0<x<1$

2. $1<x<2$

3. $2<x<3$

4. $4<x<5$

Q.

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

(i) x must be able to take each and every value of A.

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

1. Both (i) and (ii)

2. Only (i)

3. None of these

4. Only (ii)

Q.

Which of the following is correct about the function f(x)=0?

1. Its an even function

2. Its an odd function

3. Range is R and domain is R

4. Range is 0 and domain is R

Q.

How many of the following are functions ?

(i) $f\left(x\right)=x^5;\{-1,0,1\}\to \{0,1,2\}$

(ii) $f\left(x\right)=\mp \sqrt{x};\{0,4,9\}\to \{0,2,-2,3,-3\}$

(iii) $f\left(x\right)=\sqrt{x};\{0,4,9\}\to \{0,2,-2,3,-3\}$

(iv) $f\left(x\right)=x^2;\{0,1,2\}\to \{0,1,2,3,4\}$

1. (i)

2. (ii)&(iii)

3. (iii)

4. (iii) & (iv)

Q. Let $f\left(x\right)$ be a differentiable function which satisfies the equation $f\left(xy\right)=f\left(x\right)+f\left(y\right)$ for all $x>0$, $y>0$

1. $\frac{f^{\prime }\left(x\right)}{x}$
2. $\frac{1}{x}$
3. $f^{\prime }\left(x\right)$
4. $f^{\prime }\left(x\right).\left(\ln x\right)$

Q.

If y = f (x): A $\to$ B, then which of the following is true if f(x) is function in x.

(i) x must be able to take each and every value of A

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

1. Only (i)

2. None of these

3. Only (ii)

4. Both (i) and (ii)

Q.

Add $t-t^2-14;$ $15t^3+13+9t-8t^2$;$12t^2-19-24t$ and $4t-9t^2+19t^3$. [4 MARKS]

Q.

$f\left(x\right)=|x-2|andg\left(x\right)=f(f(x\left)\right)$, $x>20$, then $g\left(x\right)=$

1. $1$

2. $0$

3. $9$

4. $18$

Q.

Which of the following are the graphs of even functions?

1. 

2. 

3. 

4. 

Q.

Which of the following relations between two sets are functions?

1. 

2. 

3. 

4. 

Q.

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

1. 
2. 
3. 
4. 

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 and g cannot be determined

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

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

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

Q.

Let $f\left(x\right)=\{\begin{array}{cc}1+x,& 0\le x\le 2\\ 3-x,& 2<x\le 3\end{array}$ then $f\left\{f\right(x\left)\right\}=$

1. $\{\begin{array}{cc}2+x,& 0\le x\le 1\\ 2-x,& 1<x\le 2\\ 4-x,& 2<x\le 3\end{array}$

2. $\{\begin{array}{cc}2+x,& 0\le x\le 2\\ 4-x,& 2<x\le 3\end{array}$

3. none of these

4. $\{\begin{array}{cc}2+x,& 0\le x\le 2\\ 2-x,& 2<x\le 3\end{array}$

Q.

The relation f is defined by f(x) = $\{\begin{array}{cc}{x}^2,& 0\le x\le 3\\ 3x,& 3\le x\le 10\end{array}$ and

The relation g is defined by g(x) = $\{\begin{array}{cc}{x}^2,& 0\le x\le 2\\ 3x,& 2\le x\le 10\end{array}$ Then,

1. 'f' is a function and 'g' is not a function

2. Both 'f' and 'g' are functions

3. 'g' is a function and 'f' is not a function

4. neither 'f' nor 'g' are functions

Q. A real valued function $f\left(x\right)$ satisfies the function equation $f(x-y)=f\left(x\right)f\left(y\right)-f(a-x)f(a+y)$ where a is a given constant $f\left(0\right)=1,f(2a-x)$ is equal to

1. f(a) + f(a - x)
2. -f(x)
3. f(x)
4. f(-x)

Q.

If $f:[1,\infty )\to [0,\infty )andf\left(x\right)=\frac{x}{1+x}$ then f is

1. neither one-one nor onto
2. one-one and into
3. onto but not one-one
4. one-one and onto

Q. If $f\left(x\right)f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)\forall xϵR-0$ where f(x) be a polynomial function and f(5) =126 then f(3)=

1. 26
2. 28
3. 27
4. 25

Q.

Match the conditions/expressions in Column I with statement in Column II.

Let $f_1:R\to R,f_2:[0,\infty ]\to R,f_3:R\to R$ and $f_4:R\to [0,\infty )$ be defined by $f_1\left(x\right)=\{\begin{array}{c}|x|,ifx<0\\ e^x,ifx\ge 0\end{array}{f}_2\left(x\right)=x^2;f_3\left(x\right)=\{\begin{array}{c}sinx,ifx<0\\ x,ifx\ge 0\end{array}$ and $f_4\left(x\right)=\{\begin{array}{c}{f}_2\left[f_1\right(x\left)\right],ifx<0\\ f_2\left[f_1\right(x\left)\right]-1,ifx\ge 0\end{array}$
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ a.f_{4}is& p.\text{onto but not one-one}\\ b.f_{3}is& q.\text{neither continuous nor one-one}\\ c.f_{2}of{f}_{1}is& r.\text{differentiable but not one-one}\\ d.f_{2}is& s.\text{continuous and one-one}\end{array}$

1. A-p B-r C-s D-q

2. A-r B-p C-s D-q

3. A-r B-p C-q D-s

4. A-p B-r C-q D-s

Q. If the function $f:R\to$ A given by $f\left(x\right)=\frac{x^2}{x^2+1}$ is a surjection, then A is

1. [0, 1]
2. (0, 1]
3. [0, 1)
4. R

Q.

Let $f\left(x\right)$ be a function defined on $[-1,1]$.If the distance between $(0,0)$ and $(x,f(x\left)\right)$ is 1 unit , then the function $f\left(x\right)$ may be.

1. $\pm \sqrt{(1-x^2)}$

2. $\sqrt{(1-x^2)}$

3. $-\sqrt{(1-x^2)}$

4. $\sqrt{(1+x^2)}$

Q.

Find the value of sgn(-1) + sgn(0) + sgn(1), where sgn(x) is the signum function

___.

Q.

If (x + a) is a factor of $x^2+px+q$ and $x^2+mx+n$, then the value of a is

1. $\frac{m-p}{n-q}$

2. $\frac{m+p}{n+q}$

3. $\frac{n-q}{m-p}$

4. $\frac{q-n}{p-m}$

Q.

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

1. 

2. 

3. 

4.