Functions

Statement I: $f(x)=\frac{x}{1+|x|}$ is one-one

To check if a function is one-one (injective), we can check its monotonicity (whether it's always increasing or always decreasing).

1. For $x \geq 0: f(x)=\frac{x}{1+x}$. The derivative is $f^{\prime}(x)=\frac{1}{(1+x)^2}$, which is always positive.

2. For $x<0$ : $f(x)=\frac{x}{1-x}$. The derivative is $f^{\prime}(x)=\frac{1}{(1-x)^2}$, which is also always positive.

Since the derivative is positive for all $x$ in the domain, the function is strictly increasing. A strictly monotonic function is always one-one.

Statement I is True.

Statement II: $f(x)=\frac{x^2+4 x-30}{x^2-8 x+18}$ is many-one

Discriminant of Denominator: For $x^2-8 x+18, D=(-8)^2-4(1)(18)=64- 72=-8$. Since $D<0$, the denominator is always positive and the function is continuous for all $x \in \mathbb{R}$.

Behavior at Infinity: As $x \rightarrow \infty$ and $x \rightarrow-\infty$, the value of $f(x)$ approaches 1 (the ratio of the leading coefficients).

Turning Points: Since the function starts near $y=1$ at $-\infty$, moves through various values, and returns toward $y=1$ at $+\infty$, it must have at least one local maximum or minimum. This means a horizontal line will intersect the graph at more than one point.

Also, let

$ y=\frac{x^2+4 x-30}{x^2-8 x+18} $

Cross multiply:

$ y\left(x^2-8 x+18\right)=x^2+4 x-30 $

Rearrange:

$ (y-1) x^2+(-8 y-4) x+(18 y+30)=0 $

This is a quadratic in $x$.

For many values of $y$, this quadratic has two distinct real solutions. Function is MANY-ONE

$ f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x) $

Since the trig functions change signs depending on which quadrant $x$ is in, we analyze the value of $f(x)$ quadrant by quadrant.

Given that $x \neq \frac{n \pi}{2}$, we don't have to worry about any of these terms being zero. Each Sgn term will simply be either 1 (if positive) or -1 (if negative).

Range of $f(x)$ :

From table, Range $=\{-2,0,4\}$

Sum of all elements in this range $=4+(-2)+0=2$

Given

$g(x)=3 x^2+2 x-3, f(0)=-3$, and $4 g(f(x))=3 x^2-32 x+72$

Since $g(x)$ is a quadratic function (degree 2) and the composite function $4 g(f(x))$ is also a quadratic expression $\left(3 x^2-32 x+72\right)$, $f(x)$ must be a linear function.

• Let $f(x)=a x+b$.

• Using $f(0)=-3: a(0)+b=-3 \Longrightarrow b=-3$.

• So, $f(x)=a x-3$.

First, find $g(f(x))$ by substituting $f(x)$ into $g(x)$ :

$ g(f(x))=3(f(x))^2+2 f(x)-3 $

Substitute this into the given equation $4 g(f(x))=3 x^2-32 x+72$ :

$ 4\left[3(a x-3)^2+2(a x-3)-3\right]=3 x^2-32 x+72 $

Solve for the Coefficient ' $a$ '

Expand the left side of the equation:

• $4\left[3\left(a^2 x^2-6 a x+9\right)+2 a x-6-3\right]=3 x^2-32 x+72$

• $4\left[3 a^2 x^2-18 a x+27+2 a x-9\right]=3 x^2-32 x+72$

• $4\left[3 a^2 x^2-16 a x+18\right]=3 x^2-32 x+72$

• $12 a^2 x^2-64 a x+72=3 x^2-32 x+72$

Compare the coefficients:

• For $x^2: 12 a^2=3 \Longrightarrow a^2=\frac{1}{4} \Longrightarrow a= \pm \frac{1}{2}$

• For $x$ : $-64 a=-32 \Longrightarrow a=\frac{32}{64}=\frac{1}{2}$ Thus, $f(x)=\frac{1}{2} x-3$. Calculate $f(g(2))$

Find $g(2)$

$ g(2)=3(2)^2+2(2)-3=12+4-3=13 $

Step B: Find $f(g(2))$ which is $f(13)$

$ \begin{aligned} &f(13)=\frac{1}{2}(13)-3=6.5-3=3.5=\frac{7}{2} \end{aligned} $

Given $m=\sum_{i=1}^9(i)^2=1^2+2^2+\cdots 9$

We know, $1^2+2^2+3^2+\cdots n^2=\frac{n(n+1)(2 n+1)}{6}$

so, $m=\frac{9 \times 10 \times 19}{6}=3 \times 5 \times 19=285$

Given equation

$ 3 f(x)+2 f\left(\frac{m}{19 x}\right)=5 x, x \neq 0 $

put $\mathrm{m}=285,3 f(x)+2 f\left(\frac{285}{19 x}\right)=5 x$

$ 3 f(x)+2 f\left(\frac{15}{x}\right)=5 x-(1) $

put $x=\frac{15}{x}$

$ \begin{aligned} & 3 f\left(\frac{15}{x}\right)+2 f\left(\frac{15}{15 / x}\right)=5 \times \frac{15}{x} \\ & 3 f\left(\frac{15}{x}\right)+2 f(x)=\frac{75}{x}-(2) \end{aligned} $

from (1) $f\left(\frac{15}{x}\right)=\frac{5 x-3 f(x)}{2}$ substitute in equation (2)

$ \begin{aligned} & 3\left[\frac{5 x-3 f(x)}{2}\right]+2 f(x)=\frac{75}{x} \\ & 15 x-9 f(x)+4 f(x)=\frac{150}{x} \\ & 5 f(x)=15 x-\frac{150}{x} \\ & f(x)=3 x-\frac{30}{x} \end{aligned} $

to find $f(5)-f(2)$

$ \begin{aligned} & f(5)-f(2)=\left(3(5)-\frac{30}{5}\right)-\left(3 \times 2-\frac{30}{2}\right) \\ & =(15-6)-(6-15) \end{aligned} $

= $9-(-9)=18$

$ \begin{aligned} & \text { (3) } \Rightarrow f(x)=0 \Rightarrow[x]=3 \text { or }[x]=-2 \\ & \Rightarrow x \in[3,4) \cup[-2,-1) \Rightarrow f(x)=0 \text { has infinite solutions } \\ & f(x)=[x]^2-[x+3]-3 \\ & =\left[x^2\right]-[x]-6=([x]-3)([x]+2) \end{aligned} $

$ \begin{aligned} & (2) \Rightarrow \int_0^2 f(x) d x=\int_0^2\left([x]^2-[x]-6\right) d x \\ & \int_0^2 f(x) d x=\int_0^2\left([x]^2-[x]-6\right) d x=\int_0^2\left([x]^2-[x]\right) d x-12=-12 \\ & (4) \Rightarrow f(x)<0 \Rightarrow([x]-3)([x]+2)<0 \Rightarrow-2<[x]<3 \Rightarrow-1 \leq[x]<3 \\ & (1) \Rightarrow f(x)>0 \Rightarrow[x]<-2 \text { or }[x]>3 \Rightarrow x<-2 \text { or } x \geq 4 \Rightarrow x \in(-x,-2) 0[4, \infty] \end{aligned} $

$ \begin{aligned} & \text { let } x^2-10 x+85=\lambda \\ & \therefore \text { Domain for first term } \lambda>0 \ldots \ldots \ldots \ldots .(1) \\ & \& 7-\log _2 \lambda > 0 \Rightarrow \lambda < 2^7 \ldots \ldots \ldots .(2) \\ & \& \log _5\left(7-\log _2 \lambda\right) > 0 \Rightarrow \lambda < 2^6 \ldots \ldots . .(3) \\ & \therefore \text { from }(1),(2) \&(3) \\ & 0 < \lambda < 2^6 \\ & 0 < x^2-10 x+85 < 64 \Rightarrow x \in(3,7) \ldots \ldots . .(A) \\ & \& \text { domain for second term }-1 \leq \frac{3 x-7}{x-17} \leq 1 \Rightarrow x \in[-5,6] \ldots \ldots \ldots . . \end{aligned} $

From (A) \& (B), domain of function will be ( 3,6 ]

$ \Rightarrow \alpha=3, \beta=6 \Rightarrow \alpha+\beta=9 $

$ \begin{aligned} & f(x+y)=f(x) \cdot f(y) \Rightarrow f(x)=a^x \\ & \left(\because f(1)=7 \Rightarrow a^1=7\right) \end{aligned} $

So $f(x)=7^x$

Now

$ g(x+y)=g(x y)(\text { put } y=1) \Rightarrow g(x+1)=g(x) $

So $g(1)=g(2)=g(3)=\ldots \ldots \ldots=g(n)=1$

$ \begin{aligned} & \text { Given } \sum_{x=1}^n \frac{f(x)}{g(x)}=19607 \\ & \sum_{x=1}^n \frac{7^x}{1}=19607 \Rightarrow 7\left(\frac{7^n-1}{7-1}\right)=19607 \\ & 7^n-1=\frac{6}{7} \times 19607 \\ & 7^n=16807 \Rightarrow n=5 \end{aligned} $

$ \begin{aligned} & f(x)=\sin ^{-1}\left[\frac{5-x}{3+2 x}\right]+\frac{1}{\log _e(10-x)} \\ & -1 \leq \frac{5-x}{3+2 x} \leq 1 \\ & \frac{2-3 x}{3+2 x} \leq 0 \\ & (3 x-2)(2 x+3) \geq 0 \\ & (3 x-2)(2 x+3) \geq 0 x \in\left(-\infty,-\frac{3}{2}\right] \cup\left[\frac{2}{3}, \infty\right) \\ & \frac{8+x}{3+2 x} \geq 0 \\ & x \in(-\infty,-8] \cup\left[\frac{-3}{2}, \infty\right) \\ & x \in(-\infty,-8] \cup\left[\frac{2}{3}, 10\right)-\{9\}=6(\alpha+\beta+\gamma+\delta)=70 \end{aligned} $

$\begin{aligned} & y=\frac{5-x}{x^2-3 x+2} \\ & y x^2-3 x y+2 y+x-5=0 \\ & y z^2+(-3 y+1) x+(2 y-5)=0 \end{aligned}$

Case I : If $y=0$ (Accepted)

$\Rightarrow x=5$

Case II : If $y \neq 0$

$\begin{aligned} & \mathrm{D} \geq 0 \\ & (-3 y+1)^2-4(y)(2 y-5) \geq 0 \\ & 9 y^2+1-6 y-8 y^2+20 y \geq 0 \\ & y^2+14 y+1 \geq 0 \\ & (y+7)^2-48 \geq 0 \\ & |y+7| \geq 4 \sqrt{3} \\ & \Rightarrow y+7 \geq 4 \sqrt{3} \text { or } \mathrm{y}+7 \leq-4 \sqrt{3} \\ & \Rightarrow \mathrm{y} \geq 4 \sqrt{3}-7 \text { or } \mathrm{y} \leq-4 \sqrt{3}-7 \end{aligned}$

From Case I and Case II

$y \in(-\infty,-4 \sqrt{3}-7] \cup[4 \sqrt{3}-7, \infty)$

$\begin{aligned} & \text { So } \alpha=-4 \sqrt{3}-7 \\ & \quad \beta=4 \sqrt{3}-7 \\ & \begin{aligned} \Rightarrow a^2+b^2 & =(-4 \sqrt{3}-7)^2+(4 \sqrt{3}-7)^2 \\ & =2(48+49) \\ & =194 \end{aligned} \end{aligned}$

$f(x)=\log _4\left(\log _3\left(\log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)\right)\right.$

$\begin{aligned} & \log _3\left(\log _1\left(8-\log _2\left(x^2+4 x+5\right)\right)\right)>0 \\ & \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)>1 \\ & 8-\log _2\left(x^2+4 x+5\right)>7 \\ & -\log _2\left(x^2+4 x+5\right)>-1 \\ & \log _2\left(x^2+4 x+5\right)<1 \\ & x^2+4 x+5<2 \\ & x^2+4 x+3<0 \\ & \Rightarrow(x+3)(x+1)<0 \quad \ldots(1) \\ & \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)>0 \\ & 8-\log _2\left(x^2+4 x+5\right)>1 \\ & \log _2\left(x^2+4 x+5\right)<9 \\ & x^2+4 x+5<2^9 \\ & x^2+4 x+5<512 \\ & \Rightarrow x^2+4 x-507<0 \\ & \Rightarrow x=-4 \pm \sqrt{16+2028} \end{aligned}$

$\begin{aligned} & x=\frac{-4 \pm \sqrt{2044}}{2} \quad\text{..... (2)}\\ & \Rightarrow\left(x-\left(\frac{-4+\sqrt{2044}}{2}\right)\right)\left(x-\left(\frac{-4-\sqrt{2044}}{2}\right)\right)<0 \\ & x^2+4 x+5>0 \\ & D>0 \\ & x \in R \\ & \text { Also, } 8-\log _2\left(x^2+4 x+5\right)>0 \\ & \log _2\left(x^2+4 x+5\right)<8 \\ & x^2+4 x+5<256 \\ & \Rightarrow x^2+4 x-251<0 \\ & \Rightarrow x=-4 \pm \sqrt{16+1004} \\ & \Rightarrow x=\frac{-4 \pm \sqrt{1020}}{2} \end{aligned}$

$\begin{aligned} &\Rightarrow\left(x-\left(\frac{-4+\sqrt{1020}}{2}\right)\right)\left(x-\left(\frac{-4-\sqrt{1020}}{2}\right)\right)<0\\ &\therefore \text { Intersection of (1), (2) and (3) } \end{aligned}$

$\begin{aligned} & \therefore x \in(-3,-1) \\ & -1 \leq \frac{7 x+10}{x-2} \leq 1 \\ & \Rightarrow x \in[-2,-1] \\ & \therefore \alpha^2+\beta^2+\gamma^2+\delta^2=(-3)^2+(-1)^2+(-2)^{-2}+(-1)^2 \\ & =9+1+4+1 \\ & =15 \end{aligned}$

$\begin{aligned} & g(2)=4, g(4)=\frac{10}{3} \\ & f \text { is decreasing in }\left(\frac{10}{3}, 4\right) \\ & \therefore \quad \alpha=f(4)=\frac{1}{2} \\ & \beta=f\left(\frac{10}{3}\right)=\frac{29}{56} \\ & \frac{1}{\beta-\alpha}=\frac{1}{\frac{29}{56}-\frac{1}{2}}=56 \end{aligned}$

$\begin{aligned} & f(x)+3 f\left(\frac{24}{x}\right)=4 x, x \neq 0 \quad \ldots(1)\\ & \text { replace } x \text { by } \frac{24}{x} \\ & f\left(\frac{24}{x}\right)+3 f\left(\frac{24}{24}\right)=4\left(\frac{24}{x}\right)=\frac{96}{x} \quad \ldots(2) \\ & 3 \times(2)-(1) \\ & \Rightarrow 8 f(x)=\frac{96.3}{x}-4 x \Rightarrow f(x)=\frac{36}{x}-\frac{x}{2} \\ & f(3)+f(8)=\left(12-\frac{3}{2}\right)+\left(\frac{36}{8}-4\right) \end{aligned}$

$=8+\frac{36}{8}-\frac{12}{8}=11$

$\begin{aligned} & 1-\log _4\left(x^2-9 x+18\right)>0 \\ & \log _4\left(x^2-9 x+18\right)<1 \\ & x^2-9 x+18<4 \\ & x^2-9 x+14<0 \\ & x \in(2,7) \\ & x^2-9 x+18>0 \\ & x \in(-\infty, 3) \cup(6, \infty) \end{aligned}$

$\begin{aligned} & x \in(2,3) \cup(6,7) \\ & \alpha+\beta+\gamma+\delta=18 \end{aligned}$

$\frac{2 x-3}{4 x+5}>0$

$\begin{aligned} & \therefore \quad x \in\left(-\infty,-\frac{5}{4}\right) \cup\left(\frac{3}{2}, \infty\right) ........(i) \\ & -1 \leq \frac{3 x+4}{2-x} \leq 1 \\ & \frac{3 x+4}{2-x} \leq 1 \\ & \Rightarrow \quad \frac{3 x+4}{2-x}-1 \leq 0\end{aligned}$

$\begin{aligned} & \Rightarrow \quad \frac{3 x+4-2+x}{x-2} \geq 0 \\ & \Rightarrow \quad \frac{4 x+2}{x-2} \geq 0 \\ & \Rightarrow \quad x \in\left(-\infty,-\frac{1}{2}\right] \cup(2, \infty)........(ii)\end{aligned}$

$ \begin{aligned} & \frac{3 x+4}{2-x} \geq-1 \\ \Rightarrow & \frac{3 x+4}{2-x}+1 \geq 0 \\ \Rightarrow & \frac{3 x+4+2-x}{2-x} \geq 0 \\ \Rightarrow & \frac{2 x+6}{x-2} \leq 0 \\ \therefore & x \in[-3,2)........(iii) \end{aligned} $

Taking intersection of (i), (ii) and (iii)

$ \begin{aligned} & x \in\left[-3,-\frac{5}{4}\right) \\ & \alpha=-3, \beta=-\frac{5}{4} \\ & \alpha^2+4 \beta=4 \end{aligned} $

$x+|x|= \begin{cases}2 x, & x \geq 0 \\ 0, & x<0\end{cases}$

$\Rightarrow \frac{1}{\sqrt{x+|x|}}$, domain is $x>0$, as $2 x \neq 0$

Similarly,

$\begin{aligned} &\frac{1}{\sqrt{3 x+10-x^2}} \text { is defined when } 3 x+10-x^2>0\\ &\begin{aligned} \Rightarrow & x^2-3 x-10<0 \\ & (x-5)(x+2)<0 \\ \Rightarrow & x \in(-2,5) \\ \Rightarrow & \text { Domain will be }(0, \infty) \cap(-2,5)=(0,5) \\ \Rightarrow & (1+a)^2+b^2=1+25=26 \end{aligned} \end{aligned}$

$\begin{aligned} & f_1(x)=\log _5\left(18 x-x^2-77\right) \\ & \therefore 18 x-x^2-77>0 \\ & \quad x^2-18 x+77<0 \\ & \quad x \in(7,11) \alpha=7, \beta=11 \\ & f_2(x)=\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right) \\ & \therefore \quad x-1>0, x-1 \neq 1, \frac{2 x^2+3 x-2}{x^2-3 x-4}>0 \\ & \quad x>1, x \neq 2, \frac{(2 x-1)(x+2)}{(x-4)(x+1)}>0 \\ & \quad x>1, x \neq 2, \end{aligned}$

$\begin{aligned} & \therefore \quad x \in(4, \infty) \\ & \therefore \gamma=4 \\ & \therefore \alpha^2+\beta^2+\gamma^2=49+121+16 \\ & =186 \end{aligned}$

as $f(x)$ is onto hence $A$ is range of $f(x)$

$\text { now } \begin{aligned} f^{\prime}(x) & =6 x^2-30 x+36 \\ & =6(x-2)(x-3) \end{aligned}$

$f(2)=16-60+72+7=35$

$\begin{aligned} & \mathrm{f}(3)=54-135+108+7=34 \\ & \mathrm{f}(0)=7 \end{aligned}$

hence range $\in[7,35]=\mathrm{A}$

also for range of $g(x)$

$\begin{aligned} & g(x)=1-\frac{1}{x^{2025}+1} \in[0,1)=B \\ & s=\{0,7,8, \ldots . .35\} \text { hence } n(s)=30 \end{aligned}$

$\begin{aligned} & f(x)=\frac{2^x}{2^x+\sqrt{2}} \\ & f(x)+f(1-x)=\frac{2^x}{2^x+\sqrt{2}}+\frac{2^{1-x}}{2^{1-x}+\sqrt{2}} \\ & =\frac{2^x}{2^x+\sqrt{2}}+\frac{2}{2+\sqrt{2} 2^x}=\frac{2^x+\sqrt{2}}{2^x+\sqrt{2}}=1 \end{aligned}$

$\begin{array}{r} \text { Now, } \sum_{\mathrm{k}=1}^{81} \mathrm{f}\left(\frac{\mathrm{k}}{82}\right)=\mathrm{f}\left(\frac{1}{82}\right)+\mathrm{f}\left(\frac{2}{82}\right)+\ldots \ldots+\mathrm{f}\left(\frac{81}{82}\right) \\ =\mathrm{f}\left(\frac{1}{82}\right)+\mathrm{f}\left(\frac{2}{82}\right)+\ldots \ldots .+\mathrm{f}\left(1-\frac{2}{82}\right)+\mathrm{f}\left(1-\frac{1}{82}\right) \\ =\left[\mathrm{f}\left(\frac{1}{82}\right)+\mathrm{f}\left(1-\frac{1}{82}\right)\right]+\left[\mathrm{f}\left(\frac{2}{82}\right)+\mathrm{f}\left(1-\frac{2}{82}\right)\right]+\ldots . .40 \text { cases }+\mathrm{f}\left(\frac{41}{82}\right) \end{array}$

$\begin{aligned} & =(1+1+\ldots 40 \text { times })+\frac{2^{1 / 2}}{2^{1 / 2}+2^{1 / 2}} \\ & 40+\frac{1}{2}=\frac{81}{2} \end{aligned}$

$\begin{aligned} & f(x)=(3 a+2) x^2+\left(\frac{a+2}{a-1}\right) x+b \\ & f\left(x+\frac{1}{2}\right)=f(x)+f(y)+1-\frac{2}{7} x y\quad\text{.... (1)} \end{aligned}$

In (1) Put $x=y=0 \Rightarrow f(0)=2 f(0)+1 \Rightarrow f(0)=-1$

So, $\mathrm{f}(0)=0+0+\mathrm{b}=-1 \Rightarrow \mathrm{~b}=-1$

$\begin{aligned} & \text { In (1) Put } y=-x \Rightarrow f(0)=f(x)+f(-x)+1+\frac{2}{7} x^2 \\ & -1=2(3 a+2) x^2+2 b+1+\frac{2}{7} x^2 \end{aligned}$

$\begin{aligned} & -1=\left(2(3 a+2)+\frac{2}{7}\right) x^2+1-2 \\ & \Rightarrow 6 a+4+\frac{2}{7}=0 \\ & a=-\frac{5}{7} \end{aligned}$

So $\mathrm{f}(\mathrm{x})=-\frac{1}{7} \mathrm{x}^2-\frac{3}{4} \mathrm{x}-1$

$\Rightarrow|\mathrm{f}(\mathrm{x})|=\frac{1}{28}\left|4 \mathrm{x}^2+21 \mathrm{x}+28\right|$

Now, $28 \sum_{i=1}^5|f(i)|=28(|f(1)|+|f(2)|+\ldots+|f(5)|)$

$28 \cdot \frac{1}{28} \cdot 675=675$

$\begin{aligned} & f(x)=\frac{2^{2 \mathrm{x}}-1}{2^{2 \mathrm{x}}+1} \\ & =1-\frac{2}{2^{2 \mathrm{x}}+1} \\ & \mathrm{f}^{\prime}(\mathrm{x})=\frac{2}{\left(2^{2 \mathrm{x}}+1\right)^2} \cdot 2 \cdot 2^{2 \mathrm{x}} \cdot \ln 2 \text { i.e always }+\mathrm{ve} \end{aligned}$

so $f(x)$ is $\uparrow$ function

$\begin{aligned} & \therefore \mathrm{f}(-\infty)=-1 \\ & \mathrm{f}(\infty)=1 \\ & \therefore \mathrm{f}(\mathrm{x}) \in(-1,1) \neq \text { co-domain } \end{aligned}$

so function is one-one but not onto

$\begin{aligned} & f(x)=\frac{42^x+16}{2.2^{2 x}+16.2^x+32} \\ & f(x)=\frac{2\left(2^x+4\right)}{2^{2 x}+8.2^x+16} \\ & f(x)=\frac{2}{2^x+4} \\ & f(4-x)=\frac{2^x}{2\left(2^x+4\right)} \\ & f(x)+f(4-x)=\frac{1}{2} \end{aligned}$

So, $\quad \mathrm{f}\left(\frac{1}{15}\right)+\mathrm{f}\left(\frac{59}{15}\right)=\frac{1}{2}$

$\begin{aligned} & \text { Similarly }=f\left(\frac{29}{15}\right)+f\left(\frac{31}{15}\right)=\frac{1}{2} \\ & f\left(\frac{30}{15}\right)=f(2)=\frac{2}{2^2+4}=\frac{2}{8}=\frac{1}{4} \\ & \Rightarrow 8\left(29 \times \frac{1}{2}+\frac{1}{4}\right) \end{aligned}$

Ans. 118

$\begin{aligned} & f(x)=\ln x \\ & g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1} \\ & D_g \in R \\ & D_f \in(0, \infty) \end{aligned}$

For $D_{f o g} \Rightarrow g(x)>0$

$\begin{aligned} & \frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}>0 \\ & \Rightarrow x^4-2 x^3+3 x^2-2 x+2>0 \end{aligned}$

Clearly $\mathrm{x}<0$ satisfies which are included in option (1) only.

$ \textbf{Step 1: Total Functions with } 1 \in f(A) $

Any function $ f: A \to B $ where $ A = \{1, 2, 3, 4\} $ and $ B = \{1, 4, 9, 16\} $ is defined by choosing one of the four elements of $ B $ for each element of $ A $. Thus, the total number of functions is

$ 4^4 = 256. $

To count those functions where $ 1 $ appears at least once in the set $ f(A) $, we can use the complementary counting method: subtract the functions that never use $ 1 $. If $ 1 $ is excluded, each element of $ A $ has only 3 choices (namely, $ \{4, 9, 16\} $), so the number of such functions is

$ 3^4 = 81. $

Thus, the number of functions such that $ 1 \in f(A) $ is

$ 256 - 81 = 175. $

$ \textbf{Step 2: Counting Many-One Functions} $

In this context, "many-one functions" are understood to be non-injective functions. Since an injective (one-to-one) function from $ A $ to $ B $ must be a permutation (because both sets have 4 elements), the number of one-to-one functions is

$ 4! = 24. $

It is important to note that every injective function $ f: A \to B $ has $ f(A) = B $ (a full permutation) which automatically means $ 1 \in f(A) $.

Thus, the number of many-one (non-injective) functions $ f: A \to B $ with $ 1 \in f(A) $ is found by subtracting the one-to-one functions from the total functions that include $ 1 $:

$ 175 - 24 = 151. $

$ \boxed{151} $

This detailed explanation shows that the number of many-one functions $ f: A \rightarrow B $ such that $ 1 \in f(A) $ is indeed $ 151 $.

$\begin{aligned} & F(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathbb{R} \\ & \sin 3 x+\cos 3 x \in[-\sqrt{2}, \sqrt{2}] \\ & 2+\sin 3 x+\cos 3 x \in[2-\sqrt{2}, 2+\sqrt{2}] \\ & \Rightarrow \frac{1}{2+\sin 3 x+\cos 3 x} \in\left[\frac{1}{2+\sqrt{2}}, \frac{1}{2-\sqrt{2}}\right] \\ & \Rightarrow a=\frac{1}{2+\sqrt{2}}, b=\frac{1}{2-\sqrt{2}} \\ & \alpha=\frac{a+b}{2}=\frac{\frac{1}{2+\sqrt{2}}+\frac{1}{2-\sqrt{2}}}{2} \\ & =\frac{4}{2 \times 2}=1 \\ & \beta=\sqrt{a b}=\sqrt{\left(\frac{1}{2+\sqrt{2}}\right) \times\left(\frac{1}{2-\sqrt{2}}\right)} \\ & =\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}} \\ & \Rightarrow \frac{\alpha}{\beta}=\sqrt{2} \\ \end{aligned}$

$ \begin{array}{ll} f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right) & \\ -1 \leq \frac{x-1}{2 x+3} \leq 1 & \frac{x-1}{2 x+3}+1 \geq 0 \\ \frac{x-1}{2 x+3}-1 \leq 0 & \frac{x-1+2 x+3}{2 x+3} \geq 0 \\ \frac{x-1-2 x-3}{2 x+3} \leq 0 & \frac{3 x+2}{2 x+3} \geq 0 \end{array} $

$\begin{aligned} & \frac{-x-4}{2 x+3} \leq 0 \\ & \frac{x+4}{2 x+3} \geq 0 \end{aligned}$

$\begin{aligned} & x \in(-\infty,-4] \cup\left[\frac{-2}{3}, \infty\right) \\ & \text { Domain : } R-\left(-4, \frac{-2}{3}\right) \\ & \alpha=-4 \\ & \beta=\frac{-2}{3} \\ & 12 \times \alpha \beta=12 \times 4 \times \frac{2}{3}=32 \end{aligned}$

$\begin{aligned} & f(x)=\left\{\begin{array}{l} -a \quad \text { if }-a \leq x \leq 0 \\ x+a \quad \text { if } 0< x \leq a \end{array}\right. \\ & f(|x|)=\left\{\begin{array}{cc} -a & -a \leq|x| \leq 0 \\ |x|+a & \text { if } 0 < |x| \leq a \end{array}\right. \end{aligned}$

$|x|<0$ is not possible, so

$\begin{aligned} & f(|x|)= \begin{cases}x+a & -a \leq x \leq 0 \\ -x+a & 0 < x \leq a\end{cases} \\ & |f(x)|= \begin{cases}a & -a \leq x \leq 0 \\ x+a & 0 < x \leq a\end{cases} \\ & h(x)= \begin{cases}-\frac{x}{2} & -a \leq x \leq 0 \\ 0 & 0 < x \leq a\end{cases} \end{aligned}$

Neither one-one nor onto.

$f\left(\frac{1}{\pi}\right)< f\left(\frac{1}{e}\right) \quad \text { as } \frac{1}{\pi}<\frac{1}{e}$

$\begin{aligned} & \Rightarrow\left(\frac{1}{1}\right)^{\frac{2}{\pi}}<\left(\frac{1}{\frac{1}{e}}\right)^{\frac{2}{e}} \\ & \Rightarrow(\pi)^{\frac{2}{\pi}}<(e)^{\frac{2}{e}} \\ & \Rightarrow \pi^e < e^\pi \end{aligned}$

$\begin{aligned} & f(x)=\frac{1}{7-\sin 5 x} \\\\ & -1 \leq \sin 5 x \leq 1 \\\\ & -1 \leq-\sin 5 x \leq 1 \\\\ & -1+7 \leq 7-\sin 5 x \leq 1+7 \\\\ & 6 \leq 7-\sin 5 x \leq 8 \\\\ & \frac{1}{8} \leq \frac{1}{7-\sin 5 x} \leq \frac{1}{6} \\\\ & \frac{1}{8} \leq f(x) \leq \frac{1}{6} \\\\ & \text { Range }=\left[\frac{1}{8}, \frac{1}{6}\right] \end{aligned}$

The function $ f(x)=\frac{x^2+2x-15}{x^2-4x+9}, x \in \mathbb{R} $ can be simplified to $ f(x)=\frac{(x-3)(x+5)}{x^2-4x+9} $.

For $ x=3 $ and $ x=-5 $, $ f(x) $ equals 0. Therefore, $ f(x) $ is not one-one as it yields the same output for different input values.

The range of $ f(x) $ is $ [-2, 1.6] $, indicating that $ f(x) $ does not cover all possible real values. Consequently, $ f(x) $ is not onto.

Thus, the function is neither one-one nor onto.

$\begin{aligned} & f(x)= \begin{cases}x-1, & x \geq 1 \\ 1-x & x<0\end{cases} \\ & g(x)=\left\{\begin{array}{cc} e^x & ; \quad x \geq 0 \\ x+1 & ; \quad x \leq 0 \end{array}\right. \end{aligned}$

$\begin{aligned} f(g(x)) & = \begin{cases}g(x)-1, & g(x) \geq 1 \\ 1-g(x) & g(x)<1\end{cases} \\ & =\left\{\begin{array}{cl} e^x-1 & x \geq 0 \\ -x & x<0 \end{array}\right. \end{aligned}$

$f(1)+f(3)=14$

Case I

$\begin{aligned} & f(1)=2, f(3)=12 \\ & f(1)=12, f(3)=2 \end{aligned}$

Total one-one function

$\begin{aligned} & =2 \times 5 \times 4 \times 3 \\ & =120 \end{aligned}$

Case II

$\begin{aligned} & f(1)=4, f(3)=10 \\ & f(1)=10, f(3)=4 \end{aligned}$

Total one-one function

$\begin{aligned} & =2 \times 5 \times 4 \times 3 \\ & =120 \\ & \text { Total cases }=120+120=240 \end{aligned}$

To find the domain of the function $f(x) = \frac{\sqrt{x^2-25}}{(4-x^2)}+\log_{10}(x^2+2x-15),$ we need to consider the domain conditions for both the square root function and the logarithmic function.

The square root function $\sqrt{x^2-25}$ requires that the argument of the square root be non-negative, so $x^2 - 25 \geq 0.$ This inequality is satisfied when $x \leq -5 \quad \text{or} \quad x \geq 5.$

The denominator of the rational part of $f(x)$, $(4-x^2)$, cannot be zero, otherwise, the function will become undefined due to division by zero. Thus, we must have $4 - x^2 \neq 0.$ This inequality is violated when $x = \pm2.$

Combining these conditions gives us the domain for the rational part of the function: $x \in (-\infty, -5] \cup (5, \infty) \quad \text{and} \quad x \neq 2,-2.$

Moving on to the logarithmic function, $\log_{10}(x^2+2x-15)$, the argument must be positive: $x^2 + 2x - 15 > 0.$ This is a quadratic inequality, which we can factor to find the solution: $(x+5)(x-3) > 0.$ From this, we see that the inequality is satisfied for $x < -5 \quad \text{or} \quad x > 3.$

The overall domain of $f(x)$ is the intersection of the domains for each piece. Taking the intersection of the two sets gives us: $x \in (-\infty, -5) \cup (5, \infty),$

Since the question states that the domain is of the form $(-\infty, \alpha) \cup [\beta, \infty)$, we can infer that $\alpha = -5 \quad \text{and} \quad \beta = 5.$

We calculate $\alpha^2 + \beta^3$ as follows: $\alpha^2 + \beta^3 = (-5)^2 + 5^3 = 25 + 125 = 150.$

So the correct answer, representing the sum of $\alpha^2$ and $\beta^3$, is: Option D $150$.

To find $(g \circ g \circ g)(4),$ we first need to understand the composition of $f$ with itself, i.e., $(f \circ f)(x) = f(f(x)) = g(x).$ We can then repeatedly apply $g$ to get the given expression.

First, let's calculate $(f \circ f)(x) = g(x):$

$g(x) = (f \circ f)(x) = f(f(x))$

$= f\left(\frac{4x+3}{6x-4}\right)$

To evaluate this expression, we substitute $\frac{4x+3}{6x-4}$ for $x$ in the function $f(x):$

$g(x) = f\left(\frac{4x+3}{6x-4}\right) = \frac{4\left(\frac{4x+3}{6x-4}\right) + 3}{6\left(\frac{4x+3}{6x-4}\right) - 4}$

Now, we simplify the expression:

$g(x) = \frac{4(4x+3) + 3(6x-4)}{6(4x+3) - 4(6x-4)}$

$= \frac{16x + 12 + 18x - 12}{24x + 18 - 24x + 16}$

$= \frac{34x}{34}$

$= x$

So, $g(x) = x$ for all $x$ in the domain of $g$, which is $\mathbb{R}-\left\{\frac{2}{3}\right\}$. It's important to note that the domain restriction is preserved through the composition because $f(x)$ has a vertical asymptote at $x = \frac{2}{3}$ which doesn't intersect the graph.

So, $g(x)$ is the identity function on its domain, which means that applying $g$ any number of times will result in the same input for $x$ in the given domain. Hence, we have:

$ (g \circ g \circ g \circ g)(4) = g(g(g(g(4)))) = g(g(g(4))) = g(g(4)) = g(4) = 4 $

This corresponds to option D, which is $4$.

$\begin{aligned} & \frac{2 x+3}{4 x^2+x-3}>0 \text { and }-1 \leq \frac{2 x-1}{x+2} \leq 1 \\ & \frac{2 x+3}{(4 x-3)(x+1)}>0 \quad \frac{3 x+1}{x+2} \geq 0 \& \frac{x-3}{x+2} \leq 0 \end{aligned}$

$(-\infty,-2) \cup\left[\frac{-1}{3}, \infty\right)$ ..... (1)

$(-2,3]$ ..... (2)

$\left[\frac{-1}{3}, 3\right]$ ..... (3) $\quad (1) \cap(2) \cap(3)$

$\begin{aligned} & \left(\frac{3}{4}, 3\right] \\ & \alpha=\frac{3}{4} \beta=3 \\ & 5 \beta-4 \alpha=15-3=12 \end{aligned}$

$\begin{aligned} & -1 \leq\left|\frac{2-|x|}{4}\right| \leq 1 \\ & \Rightarrow\left|\frac{2-|x|}{4}\right| \leq 1 \\ & -4 \leq 2-|x| \leq 4 \\ & -6 \leq-|x| \leq 2 \\ & -2 \leq|x| \leq 6 \\ & |x| \leq 6 \end{aligned}$

$\Rightarrow x \in[-6,6]$ .... (1)

Now, $3-x\ne 1$

And $x\ne2$ .... (2)

and $3-x>0$

$x<3$ .... (3)

$\begin{aligned} & \text { From (1), (2) and (3) } \\ & \Rightarrow x \in[-6,3)-\{2\} \\ & \alpha=6 \\ & \beta=3 \\ & \gamma=2 \\ & \alpha+\beta+\gamma=11 \end{aligned}$

$f(g(x)) = \left\{ {\matrix{ {2 + 2g(x),} & { - 1 \le g(x) < 0} & {.....(1)} \cr {1 - {{g(x)} \over 3},} & {0 \le g(x) \le 3} & {.....(2)} \cr } } \right.$

$\text { By (1) } x \in \phi$

And by (2) $x \in[-3,0]$ and $x \in[0,1]$

Range of $\mathrm{f(g(x))}$ is $[0,1]$

$\begin{aligned} & f(x)=\frac{2 x+3}{2 x+1} ; x \neq-\frac{1}{2} \\ & g(x)=\frac{|x|+1}{2 x+5}, x \neq-\frac{5}{2} \end{aligned}$

Domain of $f(g(x))$

$f(g(x))=\frac{2 g(x)+3}{2 g(x)+1}$

$x \neq-\frac{5}{2}$ and $\frac{|x|+1}{2 x+5} \neq-\frac{1}{2}$

$x \in R-\left\{-\frac{5}{2}\right\}$ and $x \in R$

$\therefore$ Domain will be $\mathrm{R}-\left\{-\frac{5}{2}\right\}$

$\begin{aligned} & \mathrm{f}: \mathrm{N}-\{1\} \rightarrow \mathrm{N} \\ & \mathrm{f}(\mathrm{n})=\text { The highest prime factor of } \mathrm{n} . \\ & \mathrm{f}(2)=2 \\ & \mathrm{f}(4)=2 \\ & \Rightarrow \text { many one } \\ & 4 \text { is not image of any element } \\ & \Rightarrow \text { into } \end{aligned}$

Hence many one and into

Neither one-one nor onto.

First, the function $f(x) = \sin^{-1}(\log_{3x}(\frac{6 + 2 \log_3{x}}{-5x}))$ has several restrictions :

1. Since the arcsine function $\sin^{-1}(x)$ is only defined for $-1\leq x\leq 1$, this means that $\log_{3x}(\frac{6 + 2 \log _3 x}{-5 x})$ must be between -1 and 1.

2. For the logarithm to be defined, $3x > 0 \Rightarrow x > 0$ .......(1)
   
   because the base of a logarithm must be greater than 0 and not equal to 1. Also, $x \neq \frac{1}{3}$ .......(2)
   
   as the base cannot be 1.

3. Moreover, the inner function of the logarithm $\frac{6 + 2 \log_3{x}}{-5x}$ must be greater than 0.
   
   $ \Rightarrow $ $6+2 \log _3 x<0 \quad(\because x>0)$
   
   $ \begin{aligned} \Rightarrow & \log _3 x<-3 \\\\ \Rightarrow & x<3^{-3} \\\\ \Rightarrow & x<\frac{1}{27} .........(3) \end{aligned} $
   
   $ \Rightarrow \text { From (1), (2), (3), } 0 < x < \frac{1}{27} $

The next step is to solve the inequality for $-1 \leq \log_{3x}(\frac{6+2 \log _3 x}{-5 x}) \leq 1$. To do this, we make the observation that for the logarithmic part to be within $[-1,1]$, it must be true that $3x \leq \frac{6+2 \log_3 x}{-5x} \leq \frac{1}{3x}$.

Solving the inequality $15x^2 + 6 + 2 \log_3 x \geq 0$, we find that $x \in(0, \frac{1}{27})$ ........(4)

Likewise, solving the inequality $6+2 \log_3 x + \frac{5}{3} \geq 0$, we find that $x \geq 3^{-\frac{23}{6}}$ ........(5)

Combining Equations (3), (4), and (5), the intersection of all these intervals is $x \in[3^{-\frac{23}{6}}, \frac{1}{27})$.

Now, consider the function $g(x) = x - [x]$, where $[x]$ is the greatest integer function. For this function, the range is the fractional part of $x$. In this case, the range $(\alpha, \beta)$ is given by the minimum and maximum possible values of $x$ in its domain. Hence, $\alpha = 3^{-\frac{23}{6}}$ and $\beta = \frac{1}{27}$.

Finally, substitute these values into the equation $\alpha^{2}+\frac{5}{\beta}$:

$\alpha^{2}+\frac{5}{\beta} = (3^{-\frac{23}{6}})^2 + \frac{5}{\frac{1}{27}} = 3^{-\frac{23}{3}} + 135$ = 135.0002198.

Since $3^{-\frac{23}{3}}$ = 0.0002198 is an extremely small number, it's approximately 0. So, $\alpha^{2}+\frac{5}{\beta} \approx 135$.