Functions PYQs - Last 10 Years

Let $g(x)=1+x-[x]$ and ${ }^{\prime}$
$$ f(x)= \begin{cases}-1, & x<0 \\ 0, & x=0,[x] \text { denotes the greatest integer less } \\ 1, & x>0\end{cases} $$
than or equal to $x$. Then for all $x, f(g(x))=$

If $f: R \rightarrow A$, defined by $f(x)=\cos x+\sqrt{3} \sin x-1$ is an onto function then $A=$

The domain of the real valued function $f(x)=\frac{3}{4-x^2}+\log _{10}\left(x^3-x\right)$ is

Let $f(x)=x^2+2 b x+2 c^2$ and $g(x)=-x^2-2 c x+b^2 . x \in R$. If $b$ and $c$ are non-zero real numbers such that min $f(x)>\max g(x)$, then $\left|\frac{c}{b}\right|$ lies in the interval

A real valued function $f: A \rightarrow B$ defined by $f(x)=\frac{4-x^2}{4+x^2} \forall x \in A$ is a bijection. If $-4 \in A$, then $A \cap B=$

If the range of the function $f(x)=-3 x-3$ is $\{3,-6,-9,-18\}$, then which one of the following is not in the domain of $f$ ?

Let [ $x$ ] represent the greatest integer less than or equal to $x,\{x\}=x-[x] \sqrt{2}=1.414$ and $\sqrt{3}=1.732$. If $f(x)=\left\{x+\left[\frac{x}{1+x^2}\right]\right\}$ is a real valued function, then $f(\sqrt{2})+f(-\sqrt{3})=$

\(\text { Consider the following statements. }\)
$$ \begin{array}{cl} \hline \text { Statement I } & \begin{array}{l} \text { A function } f: A \rightarrow B \text { is said to be one-one if and } \\ \text { only if } f(x) \neq f(y) \Righ...

$[t]$ denotes the greatest integer function and $[t-m]=[t]-m$ when $m \in Z$.
If $k=2[2 x-1]-1$ and $3[2 x-2]+1=2[2 x-1]-1$, then the range of $f(x)=[k+5 x]$ is

If $f(x)=(x+1)^2-1, x \geq-1$, then $\left\{x \mid f(x)=f^{-1}(x)\right\}$ is

The interval in which the curve represented by $f(x)=2 x+\log \left(\frac{x}{2+x}\right)$ is

The sum of the least positive integer and the greatest negative integer in the range of the function $f(x)=\frac{x^2-5 x+7}{x^2-5 x-7}$ is

If a real valued function $f:[-1,2] \rightarrow B$ defined by
$$ f(x)= \begin{cases}1-x, & \text { when }-1 \leq x \leq 1 \\ x-1, & \text { when } 1 < x \leq 2\end{cases} $$
is a surjection, then $B=$

Let $f: N \rightarrow N$ be a function such that $f(x+y)=f(x)+f(y)+x y$ for every $x, y \in N$. If $f(\mathbb{l})=2$, then $\sum_{k=0}^{10} f(k)=$

The set of all real values of $x$ for which $f(x)=\sqrt{\frac{|x|-2}{|x|-3}}$ is a well defined function is

If a function $f: Z \rightarrow Z$ is defined by $f(x)=x-(-1)^x$, then $f(x)$ is

Domain of the real valued function $f(x)=\log \left(x^2-1\right)+x \operatorname{coth}^{-1} x$ is

The set of real values of $x$ such that $f(x)=\sqrt{\frac{[x]-1}{\left.[x]^2-[x]-6\right]}}$ is a real valued function is

The domain and range of a real valued function $f(x)=\cos x-3$ are respectively

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are two functions defined by $f(x)=2 x-3$ and $g(x)=5 x^2-2$, then the least value of the function $(g \circ f)(x)$ is

If $A \subseteq Z$ and the function $f: A \rightarrow R$ is defined by $f(x)=\frac{1}{\sqrt{64-(0.5)^{24+x-x^2}}}$, then the sum of all absolute value of elements of $A$ is

Which of the following function are odd?
I. $f(x)=x\left(\frac{e^x-1}{e^x+1}\right)$
II. $f(x)=k^x+k^{-x}+\cos x$
III. $f(x)=\log \left(x+\sqrt{x^2+1}\right)$

The domain of the real valued function $f(x)=\sqrt{9-\sqrt{x^2-144}}$ is

Define the function, $f, g$ and $h$ from $R$ to $R$ such that $f(x)=x^2-1, g(x)=\sqrt{x^2+1}$ and $h(x)= \begin{cases}0, \text { if } & x \leq 0 \\ x, \text { if } & x \geq 0\end{cases}$ consider the following statements
(i) fog is invertib...

The range of the real valued function $f(x)=\frac{15}{3 \sin x+4 \cos x+10}$ is

The range of the real valued function $f(x)=\left(\frac{x^2+2 x-15}{2 x^2+13 x+15}\right)$ is

The domain of the real valued function $f(x)$ $=\log _2 \log _3 \log _5\left(x^2-5 x+11\right)$ is

The real valued function $f: R \rightarrow\left[\frac{5}{2}, \infty\right)$ defined by $f(x)=|2 x+1|+|x-2|$ is

If $3 f(x)-2 f(1 / x)=x$, then $f(2)=$

Let $a > 1$ and $0 < \mathrm{b} < 1$. If $f: R \rightarrow[0,1]$ is defined by $f(x)=\left\{\begin{array}{ll}a^x, & -\infty < x < 0 \\ b^x, & 0 \leq x < \infty\end{array}\right.$, then $f(x)$ is

Let $f(x)=3+2 x$ and $g_n(x)=(f \circ f \circ f o \ldots$ in times $)(x)$, $\forall n \in N$ if all the lines $y=g_n(x)$ pass through a fixed point $(\alpha, \beta)$, then $\alpha+\beta=$

If $P(x)=x^5+a x^4+b x^3+c x^2+d x+e$ is a polynomial such that $P(0)=1, P(1)=2, P(2)=5, P(3)=10$ and $P(4)=17$, then $P(5)=$

The domain of the real valued function $f(x)=\sqrt{2+x}+\sqrt{3-x}$ is

$f: R \rightarrow R$ is defined by $f(x+y)=f(x)+12 y, \forall x, y \in R$. If $f(1)=6$, then $\sum_{r=1}^n f(r)=$

If a real valued function $f:[a, \infty) \rightarrow[b, \infty)$ defined by $f(x)=2 x^2-3 x+5$ is a bijection. Then, $3 a+2 b=$

The domain of the real valued function $f(x)=\frac{1}{\sqrt{\log _{0.5}(2 x-3)}}+\sqrt{4-9 x^2}$ is

$f(x+h)=0$ represents the transformed equation of the equation $f(x)=x^4+2 x^3-19 x^2-8 x+60=0$. If this transformation removes the term containing $x^3$ from $f(x)=0$, then $h=$

For real values of $ x $ and $ a $, if the expression $ \frac{x^3 - 3x^2 - 3x + 1}{2x^2 - 3x + 1} $ assumes all real values, then

If $ f(x) = \sqrt{x - 1} $ and $ g(f(x)) = x + 2x^2 + 1 $, then $ g(x) $ is

If a function $ f:R \rightarrow R $ is defined by $ f(x) = x^3 - x $, then $ f $ is

For $x \in R$ if $f(x)=\sqrt{\log _{10}\left(\frac{3-x}{x}\right)}$, then the domain of $f$ is

If $f(x)=x^3-x$ and $g(x)=\sin 2 x$, then $f\left(g\left(\frac{\pi}{12}\right)\right)=$

If a real valued function $f$ is defined by $f(x)=\frac{a x+\sqrt{a^2-x^2}}{b x}$, then $f$ is

If $f(a)=\log \left|\frac{1-a}{1+a}\right|$ for $a \neq\{-1,1\}$, then the set of values of all ' $a$ ', for which $f\left(\frac{2 a}{1+a^2}\right)>0$ is

Let \(f: R-\left\{\frac{-1}{2}\right\} \rightarrow R\) be defined by \(f(x)=\frac{x-2}{2 x+1}\). If \(\alpha\) and \(\beta\) satisfy the equation \(f(f(x))=-x\), then \(4\left(\alpha^2+\beta^2\right)=\)

\(f(x)=\log \left(\left(\frac{2 x^2-3}{x}\right)+\sqrt{\frac{4 x^4-11 x^2+9}{|x|}}\right) \text { is }\)

The range of the real valued function \(f(x)=\sqrt{\frac{x^2+2 x+8}{x^2+2 x+4}}\) is

If \(f(x)=\sqrt{2-x^2}\) and \(g(x)=\log (1-x)\) are two real valued functions, then the domain of the function \((f+g)(x)\) is

The domain of the real valued function \(f(x)=\sin \left(\log \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right.\) is

Let \(f(x)=x^3\) and \(g(x)=3^x\), then the quadratic equation whose roots are solutions of the equation \((f \circ g)(x)=(g \circ f)(x)\) (for \(x \neq 0\)) is