Myanmar Matriculation (Function 1, 5)

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Group (1)

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1(2011) Let the function $f: R \rightarrow R$ be defined by $f(x)=2^{x}$. What are the images of $-2$ and 2? Find $a \in R$ such that $f(a)=256$. (3 marks)

2(2010) A function $f$ is defined by $f(x)=1+2 x$.Find the value of $x$ such that $(f \circ f)(x)=4 f(x)$.$\text{ (3 marks)}$

3(2010) Functions $f$ and $g$ are defined by $f(x)=2 x+p$, where $p$ is a constant, and $g(x)=4 x+6 .$ Find the value of $p$ for which $(f \circ g)(x)=(g \circ f)(x)$.$\text{ (3 marks)}$

4(2010) Functions $g$ and $h$ are defined by $g(x)=a x+10$, where $a$ is constant, and $h(x)=3 x+5$.Find the value of $a$ for which $(h \circ g)(x)=(g \circ h)(x)$.$\text{ (3 marks)}$

5(2010) Let $f: R \rightarrow R$ be defined by $f(x)=4 x+1$.Find the formula for a function $g: R \rightarrow R$ such that $(f \circ g)(x)=21-12 x$.$\text{ (3 marks)}$

6(2011) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be $f(x)=p x+5$ and $g(x)=q x-3$, where $p \neq 0$, $q \neq 0$. If $g \circ f: R \rightarrow R$ is the identity function on $R$, find the value of $p$. (3 marks)

7(2011) A function $f$ is defined by $f(x+1)=4 x+5$. Find $a \in R$ such that $f(14)=a+14$. (3 marks)

8(2011) A function $f$ is defined by $f(2 x+1)=x^{2}-3 .$ Find $a \in R$ such that $f(5)=a^{2}-8$. (3 marks)

9(2012) Given that $f(x)=3 x-4, g(x)=x^{2}-1$. Find the values of $x$ which satisfy the equation $(g \circ f)(x)=9-3 x$. (3 marks)

10(2013) Let $f: R \rightarrow R$ is defined by $f(x)=a x-4$. Given that $f(3)=5$, find $a$. Hence solve the equation $(f \circ f)(x)=f(x)$. (3 marks)

11(2013) Let the function $f: R \rightarrow R$ and $g: R \rightarrow R$ be given by $f(x)=2 x+1$ and $g(x)=x^{2}+5$. Find the value of $a \in R$ for which $(f \circ g)(a)=f(a)+22$. (3 marks)

12(2013) A function $f$ is defined, for $x \neq 0$, by $f(x)=\frac{a}{x}+1$, where $a$ is constant. Given that $6(f \circ f)(-1)+a=0$, find the possible values of $a$. (3 marks)

13(2013) If $f: x \mapsto 2 x+b$ and $g: x \mapsto 3 a-2 x$, such that $f \circ g=g \circ f$, find the relationship between $a$ and $b$. (3 marks)

14(2014) A function $f: R \rightarrow R$ is defined by $f(3 x+1)=x^{2}+1$. Find $a \in R$ such that $f(10)=a^{2}-6.$ $\qquad\mbox{ (3 marks)}$

15(2015/FC/q02) A function $f$ is defined by $f(2 x+1)=x^{2}-3$. Find $a \in R$ such that $f(5)=a^{2}-8.$ (3 marks)

16(2016/FC/q02) A function $f: x \mapsto \frac{b}{x-a}, x \neq a$ and $a>0$ is such that $(f \circ f)(x)=x$. Show that $x^{2}-a x-b=0$. (3 marks)

17(2017/Myanmar/q02) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=k x-1$, where $k$ is a constant and $g(x)=x+12 .$ Find the value of $k$ for which $$(g \circ f)(2)=(f \circ g)(2)$$ (3 marks)

18(2019/FC/q01a) Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-1$, and $\mathrm{g}(\mathrm{x})=3 \mathrm{x}+1$. Find the values of $\mathrm{x}$ which satisfy the equation $(\mathrm{g} \circ \mathrm{f})(\mathrm{x})=7 \mathrm{x}-4, \quad$ (3 marks)

19(2019/Myanmar/q01a) Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}+1$, and $\mathrm{g}(\mathrm{x})=2 \mathrm{x}^{2}-\mathrm{x}+3$. Find the values of $\mathrm{x}$ which satisfy the equation $(\mathrm{f} \circ \mathrm{g})(\mathrm{x})=4 \mathrm{x}+1$. (3 marks)


Answer Group (1)


1$f(-2)=\frac{1}{4}, f(2)=4, a=8$
2$-\frac{1}{4}$
32
45
5$5-3 x$
6$p=\frac 53$
7$a=43$
8$a=\pm 3$
9$x=\frac{1}{3}$ (or) $x=2$
10 $a=3, x=2$
11 $a=-2$ (or) 3
12 $a=-2$ (or) 3
13 $a+b=0$
14$a=\pm 4$
15$a=\pm 3$
16Show
17$k=1$
18 $\dfrac 13$ or 2
19 $x=\frac 32$ or $x=1$

Group (5)

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1 ( 2010 ) A function $f$ is defined by $f: x \mapsto \frac{2 x}{x-4}, x \neq 4$.Find the non-zero value of $x$ for which $(f \circ f)(x)=f^{-1}(x)$.$\text{ (5 marks)}$

2 ( 2010 ) Functions $h$ and $g$ are defined by $g: x \mapsto \frac{x+1}{x-2}, x \neq 2, h: x \mapsto \frac{a x+3}{x}$, $x \neq 0$, find the value of $a$ for which $\left(h \circ g^{-1}\right)(4)=g^{-1}(2)$.$\text{ (5 marks)}$

3 ( 2010 ) A functions $f$ is defined by $f(x)=a x+1 .$ If $f^{-1}(3)=1$, find the value of $a$ and hence show that $(f \circ f)^{-1}(x)=\left(f^{-1} \circ f^{-1}\right)(x)$. $\text{ (5 marks)}$.

4 ( 2011 ) A function $f$ is defined by $f: x \mapsto \frac{a}{x}+1, x \neq 0$, where $a$ is constant. Given that $6(f \circ f)(-1)+f^{-1}(2)=0$, find the possible values of $a$. (5 marks)

5 ( 2012 ) Functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=x+7$ and $g(x)=3 x-1 .$ Find the value of $x$ for which $\left(g^{-1} \circ f\right)(x)=\left(f^{-1} \circ g\right)(x)+8$. (5 marks)

6 ( 2012 ) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=x+7$ and $g(x)=3 x-1.$ Find $\left(f^{-1} \circ g\right)(x)$ and what is the value of $b \in R$ for which $\left(f^{-1} \circ g\right)(b)=4$. (5 marks)

7 ( 2012 ) Functions $f$ and $g$ are defined by $f: x \mapsto 2 x+1$ and $g: x \mapsto \frac{2 x+5}{3-x}, x \neq 3$. Find the values of $x$ for which $\left(f \circ g^{-1}\right)(x)=x-4$. (5 marks)

8 ( 2012 ) Let $f(x)=3 x+2$ and $g(x)=\frac{2 x-3}{x-2}, x \neq 2$. Find the formulae of $f \circ g$ and $g^{-1}$ Solve the equation $g^{-1}(x)=x$. (5 marks)

9 ( 2013 ) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f: x \mapsto 3 x-1$ and $g: x \mapsto x+7$. Find the value of $x$ for which $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)+8$. (5 marks)

10 ( 2013 ) A function $f$ is defined by $f(x)=\frac{a x-3}{x-1}$ for all $x \not= 1, f(3)=6$, find the value of a and the formula of $f^{-1}$ in simplified form. Verify also that $\left(f^{-1} \circ f\right)(x)=x$. (5 marks)

11 ( 2014 ) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=3 x-1$ and $g(x)=x+7$. Find the value of $x$ for which $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)+8$. $\qquad\mbox{ (5 marks)}$

12 ( 2014 ) A function $f: R \rightarrow R$ is defined by $f(x)=p x+2$. If $f^{-1}(11)=3$, find the value of $p$ and hence show that $(f \circ f)^{-1}(x)=\left(f^{-1} \circ f^{-1}\right)(x)$. (5 marks)}

13 ( (2017/Myanmar/q07a) ) Functions $f$ and $g$ are defined by $f: x \mapsto \frac{x}{x-3}, x \neq 3, g: x \mapsto 3 x+5$. Find the value of $x$, for which $(f \circ g)^{-1}(x)=\frac{5}{3}$. (5 marks)

14 ( (2019/FC/q06a) ) Functions $\mathrm{f}: \mathrm{R} \mapsto \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \mapsto \mathrm{R}$ are defined by $\mathrm{f}(\mathrm{x})=3 \mathrm{x}-1$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}+2$. Find the value of $x$ for which $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)-4.$ (5 marks)

15 ( 2013 ) A function is defined by $f(x)=\frac{1}{3-2 x}$ for all values of $x$ except $x=\frac{3}{2}$. Find the values of $x$ which map on to themselves under the function $f$. Find also an expression for $f^{-1}$ and the value of $(f \circ f)(2)$. (5 marks)




Answer Group (5)

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1 6
2 4
3 2
4 $a=-2$ (or) 3
5 $x=1$
6 $\left(f^{-1} \circ g\right)(x)=3 x-8 ; b=4$
7 $x=0$ (or) $x=9$
8 $(f \circ g)(x)=\frac{8 x-13}{x-2}, x \neq 2, g^{-1}(x)=\frac{2 x-3}{x-2}, x \neq 2, x=1$ (or) $x=3$
9 $x=1 \quad$
10 $a=5$ ; $f^{-1}(x)=\frac{x-3}{x-5}, x \neq 5$
11 $x=1$
12 $p=3$
13 $x=\frac{10}{7}$
14 $x=3$
15 $ x=\frac{1}{2}$ (or) 1 ; $ f^{-1}(x)=\frac{3 x-1}{2 x}, x \neq 0 ;$ $\frac{1}{5} $


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