$\def\mathrm{}$
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)
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| 2 | (2010) A function $f$ is defined by $f(x)=1+2 x$.Find the value of $x$ such that $(f \circ f)(x)=4 f(x)$.$\text{ (3 marks)}$
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| 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 $(f \circ g)(x)=(g \circ f)(x)$.$\text{ (3 marks)}$
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| 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 $(h \circ g)(x)=(g \circ h)(x)$.$\text{ (3 marks)}$
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| 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 $(f \circ g)(x)=21-12 x$.$\text{ (3 marks)}$
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 14 | (2014) A function $f: R \rightarrow R$ is defined by $f(3 x+1)=x^{2}+1$. Find $a \in R$ such that $f(10)=a^{2}-6.$ $\qquad\mbox{ (3 marks)}$
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 1 | $f(-2)=\frac{1}{4}, f(2)=4, a=8$
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| 2 | $-\frac{1}{4}$
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| 3 | 2
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| 4 | 5
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| 5 | $5-3 x$
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| 6 | $p=\frac 53$
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| 7 | $a=43$
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| 8 | $a=\pm 3$
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| 9 | $x=\frac{1}{3}$ (or) $x=2$
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| 10 | $a=3, x=2$
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| 11 | $a=-2$ (or) 3
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| 12 | $a=-2$ (or) 3
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| 13 | $a+b=0$
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| 14 | $a=\pm 4$
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| 15 | $a=\pm 3$
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| 16 | Show
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| 17 | $k=1$
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| 18 | $\dfrac 13$ or 2
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| 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 $(f \circ f)(x)=f^{-1}(x)$.$\text{ (5 marks)}$
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| 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 $\left(h \circ g^{-1}\right)(4)=g^{-1}(2)$.$\text{ (5 marks)}$
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| 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 $(f \circ f)^{-1}(x)=\left(f^{-1} \circ f^{-1}\right)(x)$. $\text{ (5 marks)}$.
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 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 $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)+8$. $\qquad\mbox{ (5 marks)}$
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| 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)}
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| 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)
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| 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)
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| 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
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| 2 | 4
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| 3 | 2
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| 4 | $a=-2$ (or) 3
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| 5 | $x=1$
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| 6 | $\left(f^{-1} \circ g\right)(x)=3 x-8 ; b=4$
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| 7 | $x=0$ (or) $x=9$
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| 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) $x=3$
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| 9 | $x=1 \quad$
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| 10 | $a=5$ ; $f^{-1}(x)=\frac{x-3}{x-5}, x \neq 5$
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| 11 | $x=1$
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| 12 | $p=3$
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| 13 | $x=\frac{10}{7}$
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| 14 | $x=3$
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| 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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