Functions PYQs - Last 5 Years

The domain of the function $f(x) = {}^{(10-x)}C_{(x-6)}$ is...

The domain of the function $f(x) = \sqrt{x - 1} + \sqrt{6 - x}$ is...

If $f(x) = \dfrac{2x - 1}{x + 5}$, $x \neq -5$ then $f^{-1}(x)$ is equal to

The statement having truth value 'T' from the following is

If $f(x) = x^2$ and $g(x) = [x^2]$ where $[\cdot]$ represents the greatest integer function then, $(f \circ g)\left(\dfrac{3}{2}\right) + (g \circ f)\left(\dfrac{3}{2}\right)$ is equal to ...

If $g(x) = 1 - \sqrt{x}$ and $f(g(x)) = 5 + 4\sqrt{x} + x$, then the value of $f(6)$ is...

Let $f(x) = ax + b$ and $g(x) = cx + d$. The condition $f(g(x)) = g(f(x))$ holds for all $x$ if and only if ...

The domain of the function $f(x) = \sqrt{\dfrac{x}{1+x}}$ is $\ldots$

If $f : R \to R$ and $g : R \to R$ are defined as $f(x) = 2x - |x|$ and $g(x) = 2x + |x|$, then

If $f(x) = \dfrac{x+2}{x^2-3x+1}$, then the values of $x$ for which $f(x)$ is not defined are

If $f(x) = \dfrac{1 - x}{1 + x}$, then $f(f(\cos x)) =$

If $f(x) = \dfrac{4x + 3}{6x - 4}$, $x \neq \dfrac{2}{3}$ and $(\text{fof})(x) = g(x)$ where $g: \mathbb{R} - \left\{\dfrac{2}{3}\right\} \rightarrow \mathbb{R} - \left\{\dfrac{2}{3}\right\}$, then $(g\,o\,g\,o\,g\,o\,g\,o\,g)\,(3) =$

If $e^x + e^{f(x)} = e$, then the domain of $f(x)$ is

If $f(x-y) + f(x+y) = 2f(x)f(y)$ for all $x, y \in \mathbb{R}$, then $f(x)$ is .....

The domain of the function $f(x) = e^{\sqrt{5x - 3 - 2x^2}}$ is

If $g(x)$ is the inverse function of $f(x)$, where $f(x) = \dfrac{5x+3}{4x-1}$, then $g(1) =$

The domain of the function $f(x) = \sqrt{x - 1} + \sqrt{3 - x}$ is ...

If $g(x) = x^2 + x - 2$ and $\dfrac{1}{2}(g \circ f)(x) = 2x^2 - 5x + 2$ then $f(x)$ is equal to

The function $\mathrm{f}(x)=\sec \left[\log \left(x+\sqrt{1+x^2}\right)\right]$ is________function

If $[x]^2-5[x]+6=0$, where [.] denotes the greatest integer function, then

If $\mathrm{f}(x)=\log \left(\frac{1+x}{1-x}\right)$ and $\mathrm{g}(x)=\frac{3 x+x^3}{1+3 x^2}$, then $(\mathrm{fog})(x)=$

Let $\mathrm{f}: \mathbb{R}-\{2\} \rightarrow \mathbb{R}-\{1\}$ defined by $\mathrm{f}(x)=\frac{x-3}{x-2}$ and $\mathrm{g}: \mathbb{R} \rightarrow \mathbb{R}$ defined by $\mathrm{g}(x)=3 x-2$, then sum of all values of $x$ for which $\mathr...

If $\mathrm{f}(x)=3 x+10, \mathrm{~g}(x)=x^2-1$, then $(\mathrm{fog})^{-1}(x)=$

$$ \begin{aligned} & f(x)=\left\{\begin{array}{ll} 3-x, & -1 \leqslant x<0 \\ 1+\frac{5 x}{3}, & -3 \leqslant x \leqslant 2 \end{array}\right. \text { and } \\ & g(x)=\left\{\begin{aligned} -x, & -2 \leqslant x \leqslant 3 \\ x, & 0 \leqsla...

For a real number $x,[x]$ denotes the greatest integer less than or equal to $x$. Then the value of
$$ \begin{array}{r} {\left[\frac{1}{2}\right]+\left[\frac{1}{2}+\frac{1}{100}\right]+\left[\frac{1}{2}+\frac{2}{100}\right]+\left[\frac{1}{2...

The values of $b$ and $c$ for which the identity $\mathrm{f}(x+1)-\mathrm{f}(x)=8 x+3$ is satisfied, where $\mathrm{f}(x)=\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}$, are

The function defined by $\mathrm{f}(x)=\frac{2 x+3}{3 x+4}, x \neq-\frac{4}{3}$ is

The domain of definition of the function $f(x)$ given by the equation $2^x+2^y=2$ is

If $\mathrm{f}(x)=\frac{x}{2-x}, \mathrm{~g}(x)=\frac{x+1}{x+2}$, then (gogof) $(x)=$

If $g(x)=x^2+x-1$ and (gof) $(x)=4 x^2-10 x+5$, then $\mathrm{f}(2)$ is equal to

The domain of definition of $\mathrm{f}(x)=\frac{\log _2(x+3)}{x^2+3 x+2}$ is

For a suitable chosen real constant a, let a function $f: \mathbb{R}-\{-\mathrm{a}\} \rightarrow \mathbb{R}$ be defined by $f(x)=\frac{a-x}{a+x}$. Further suppose that for any real number $x \neq-\mathrm{a}$ and $\mathrm{f}(x) \neq-\mathrm{...

Range of the function $\mathrm{f}(x)=\frac{x^2+x+2}{x^2+x+1}, x \in \mathbb{R}$ is

Let $\mathrm{f}(x)=(x+1)^2-1, x \geqslant-1$, then the set $\left\{x / f(x)=f^{-1}(x)\right\}$ is

If $[x]^2-5[x]+6=0$, where $[\cdot]$ denotes the greatest integer function, then

If $[x]^2-5[x]+6=0$, where $[x]$ denotes the greatest integer function, then

If $\mathrm{f}:[1, \infty) \rightarrow[2, \infty)$ is given by $\mathrm{f}(x)=x+\frac{1}{x}$ then $\mathrm{f}^{-1}(x)$ equals

The domain of definition of the function $y(x)$ is given by the equation $2^x+2^y=2$, is

Let $\mathrm{f}(x)=\frac{a x}{x+1}, x \neq-1$, then for $\alpha=$ ________, $\mathrm{f}(\mathrm{f}(x))=x$.

\(\mathrm{f}: \mathbb{R}-\left(-\frac{3}{5}\right) \rightarrow \mathbb{R}\) is defined by \(f(x)=\frac{3 x-2}{5 x+3}\), then \(f \circ f(1)\) is

The function \(\mathrm{f}(x)=[x] \cdot \cos \left(\frac{2 x-1}{2}\right) \pi\), where \([\cdot]\) denotes the greatest integer function, is discontinuous at

The range of the function \(\mathrm{f}(x)=\frac{x^2}{x^2+1}\) is

If \(\mathrm{f}(x)=\frac{2 x-3}{3 x-4}, x \neq \frac{4}{3}\), then the value of \(\mathrm{f}^{-1}(x)\) is

If \(3 \mathrm{f}(x)-\mathrm{f}\left(\frac{1}{x}\right)=8 \log _2 x^3, x>0\), then \(\mathrm{f}(2), \mathrm{f}(4)\), \(f(8)\) are in

The domain of the definition of the function \(y(x)\) is given by the equation \(2^x+2^y=2\) is

If \(\mathrm{g}(x)=1+\sqrt{x}\) and \(\mathrm{f}(\mathrm{g}(x))=3+2 \sqrt{x}+x\) then \(\mathrm{f}(\mathrm{f}(x))\) is

If \(\mathrm{f}(x)=x^2+1\) and \(\mathrm{g}(x)=\frac{1}{x}\), then the value of \(\mathrm{f}(\mathrm{g}(\mathrm{g}(\mathrm{f}(x))))\) at \(x=1\) is

The domain of the function given by \(2^x+2^y=2\) is

If \(\mathrm{f}(x)=\frac{3 x+4}{5 x-7}\) and \(\mathrm{g}(x)=\frac{7 x+4}{5 x-3}\), then \(\mathrm{f}(\mathrm{g}(x))=\)

\(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} ; \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are two functions such that \(\mathrm{f}(x)=2 x-3, \mathrm{~g}(x)=x^3+5\), then \((\mathrm{fog})^{-1}(-9)\) is