# Myanmar Matriculation (Function 3, 7)

### Group (3)

 $\quad$ 1 ( 2014 ) Let $N$ be the set of natural numbers. A function $f$ from $N$ to $N$ is given by, $f(x)=$ the sum of all factors of $x$. If $f(16)=8 p-9$, then find $f\left(p^{2}\right).$ $\qquad\mbox{ (3 marks)}$ 2 ( 2012 ) A function $f$ is defined by $f: x \mapsto \frac{x+4}{2 x-1}, x \neq \frac{1}{2}$. Find the value of $p$ if $f\left(\frac{1}{p}\right)=p$. (3 marks) 3 ( 2013 ) A function $f$ from $A$ to $A$, where $A$ is the set of positive integers, is given by $f(x)=$ the sum of all positive divisors of $x$. Find the value of $k$, if $f(15)=3 k+6$. (3 marks) 4 ( 2011 ) Let the function $f: R \rightarrow R$ be given by $f(x)=c x+d$, where $c$ and $d$ are fixed real numbers. If $f(0)=-3$ and $f(2)=1$, find $c$ and $d$, and then find $f(9)$. (3 marks) 5 ( 2012 ) Let $f: R \rightarrow R$ be given by $f(x) =\frac{x+a}{x-2}, x \neq 2, f(8)=3$. Find the value of $a$ and $f^{-1}(7)$. (3 marks) 6 ( 2012 ) A function $f$ is defined by $f(x)=3 x-5$. Find the formula of $f^{-1}$. Find also the value of $k$, such that $f(k)=f^{-1}(k)$. (3 marks) 7 ( 2014 ) A function $f$ is defined by $f: x \mapsto \frac{3-x}{2 x}, x \neq 0$. Find the value of $x$ for which $f(x)=f^{-1}(x).$ $\qquad\mbox{ (3 marks)}$ 8 ( 2014 ) The function $f$ is given by $f(x)=\frac{4 x-9}{x-2}, x \neq 2$. Find the value of $x$ for which $4 f^{-1}(x)=x.$ $\qquad\mbox{ (3 marks)}$ 9 ( (2016/Myanmar/q02) ) The function $f$ is defined, for $x \in R$, by $f(x)=2 x-3$. Find the value of $x$ for which $f(x)=f^{-1}(x)$. (3 marks) 10 ( 2010 ) Given that $f(x)=\frac{x+a}{x-3}, x \neq 3$, and $f(8)=3$, find the value of $a$ and $f^{-1}(11)$.$\text{ (3 marks)}$ 11 ( 2011 ) Let $f: R \rightarrow R$ be given by $f(x)=\frac{4 x+5}{a x-1}, x \neq \frac{1}{a}, f^{-1}(3)=1$, find $a$. (3 marks) 12 ( 2012 ) A function $f$ is defined by $f(x)=\frac{x}{a}+a$. If $f^{-1}(3)=2$, find the values of $a$. (3 marks) 13 ( 2012 ) A function $f$ is such that $f(x)=\frac{2}{k x+3}$ for all $x \neq-\frac{3}{k}$ where $k \neq 0$. If $f(-1)=2$, find the value of $k$ and the formula of $f^{-1}$. (3 marks) 14 ( 2010 ) A function $f: R \rightarrow R$ is defined by $f(x)=\frac{a x-9}{x-1}, x \neq 1$.If $f^{-1}(-1)=6$, find the value of $a$ and evaluate the image of 3 under $f$.$\text{ (3 marks)}$ 15 ( 2010 ) A function $f$ is defined by $f(x)=\frac{5 x+3}{x-4}$ where $x \neq 4$.Find the formula of $f^{-1}$.$\text{ (3 marks)}$ 16 ( 2014 ) Let $f(x)=\frac{3 x}{x-4}, x \neq 4$. Find the formula of $f^{-1}$. $\qquad\mbox{ (3 marks)}$ 17 ( 2014 ) Find the formula for $f^{-1}$, the inverse function of $f$ defined by $f(x)=\frac{2}{3-4 x}$. State the suitable domain of $f.$ $\qquad\mbox{ (3 marks)}$ 18 ( 2014 ) Let the mapping $\odot$ be defined by $(x, y) \rightarrow x \odot y=x+2 y$, where $x$ and $y$ are in $A=\{0,1,2\}.$ Is this mapping a binary operation? $\qquad\mbox{ (3 marks)}$

 $\quad$ 1 $31 \quad$ 2 $p=-1$ 3 $k=6$ 4 $c=2, d=-3, f(9)=15$ 5 $a=10 ; f^{-1}(7)=4$ 6 $f^{-1}(x)=\frac{x+5}{3};k=\frac{5}{2}$ 7 $x=-\frac{3}{2}$ (or) $x=1 \quad$ 8 $x=6$ 9 $x=3$ 10 7;4 11 $a=4$ 12 $a=1$ (or) $a=2$ 13 $k=2 ; f^{-1}(x)=\frac{2-3 x}{2 x}, x \neq 0$ 14 $\frac{2}{3} ;-\frac{7}{2}$ 15 $\frac{4 x+3}{x-5}, x \neq 5$ 16 $f^{-1}(x)=\frac{4 x}{x-3}, x \neq 3$ 17 $f^{-1}(x)=\frac{3 x-2}{4 x}, x \neq 0,\left\{x \mid x \in R, x \neq \frac{3}{4}\right\} \quad$ 18 The closure property is not satified, $\odot$ is not a binary operation.

### Group (7)

 $\quad$ 1 ( 2012 ) Functions $f$ and $g$ are defined by $f(x)=3 x+a, g(x)=-3 x + b$. Given that $(f \circ f)(4)=4$ and $g(3)=g^{-1}(3)$, find the value of $a$ and of $b$. (5 marks) 2 ( ) Functions $f$ and $g$ are defined by $f(x)=\frac{x}{2-x}, x \neq 2$ and $g(x)=a x+b$. Given that $g^{-1}(7)=3$ and $(g \circ f)(5)=-7$, calculate the value of $a$ and of $b$. (5 marks) 3 ( 2014 ) Functions $f$ and $g$ are defined by $f: x \mapsto \frac{x}{x+2}, x \neq-2$ and $g: x \mapsto p x+q$, where $p$ and $q$ are constants. Given that $g(2)=12$ and $(g \circ f)(-3)=19$, find the values of $p$ and $q$. $\qquad\mbox{ (5 marks)}$ 4 ( 2011 ) Functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=a x+b$, where $a$ and $b$ are constants, $g(x)=x+7,(g \circ f)(1)=5$ and $(f \circ g)(1)=19 .$ Find the values of $a$ and $b$ and hence find the formula for $g \circ f$. (5 marks) 5 ( 2012 ) Let $f: x \mapsto a+b x, f(2 b)=b,(f \circ f)(b)=ab.$ If $f$ is not a constant function, find formula for $f$. (5 marks) 6 ( 2012 ) The functions $f:x\mapsto ax^3+bx+30.$ Then the values $x=2$ and $x=3$ which are unchanged by the mapping. Find the value of $a$ and $b$. (5 marks) 7 ( 2014 ) $f: x \mapsto \frac{12}{a x+b}, f(0)=-3, f(2)=-6$, given. Find $a$ and $b$. Find $x$ for which $f(x)=x$. $\qquad\mbox{ (5 marks)}$ 8 ( 2014 ) A function $h$ is defined by $h: x \rightarrow \frac{x+3}{x-3}, x \neq 3$. Show that $h(3+p)+h(3-p)=2$ where $p$ is positive and find the positive number $q$ such that $h(q)=q-1$. $\qquad\mbox{ (5 marks)}$ 9 ( 2014 ) Given that $f(x)=\frac{a}{x}+1, x \neq 0$. Find the formula for $f^{-1}$, state the suitable domain of $f^{-1}$. If $f^{-1}(2)=1$, find $a$. $\qquad\mbox{ (5 marks)}$ 10 ( 2013 ) Given that $f: x \mapsto \frac{x}{p}+q, f(8)=1, f^{-1}(-2)=2$, show that $\frac{p}{2}+q^{2}=10$. (5 marks)

 $\quad$ 1 $a=-8, b=12$ 2 $a=3, b=-2 \quad$ 3 $p=7,q=-2$ 4 $a=3, b=-5,(g \circ f)(x)=3 x+2$ 5 $a=-1, b=1, f(x)=x-1$ 6 $a=1, b=-18$ 7 $a=1, b=-4, x=6$ (or) $-2$. 8 $q=5$ 9 $f^{-1}(x)=\frac{a}{x-1},\{x \mid x \neq 1, x \in R\}, a=1$ 10 Show