Exercise 2.2
Question 1
If the functions f and g are given by
$\begin{aligned}&f=\{(1,2),(3,5),(4,1)\} \\&g=\{(2,3),(5,1),(1,3)\}\end{aligned}$
then define fog and gof
Sol :
Question 2
Let $A=\{1,2,3\}, B=\{4,5\}, C=\{5,6\}$.
Let $f: A \rightarrow B, g: B \rightarrow C$ be defined by $f(1)=4, f(2)=5, f(3)=4$, $g(4)=5, g(5)=6$ Find gof : $A \rightarrow C$.
Sol :
Question 3
Let the functions f and g be given by $f=\{(1,-1),(2,-4),(3,-9)\}$ and $g=\{(-1,3),(-4,7),(-9,11)\}$. Show that gof is defined while fog is not defined, Also find gof
Sol :
Question 4
Let $f: \mid a, c, d\} \rightarrow\{a, b, e\}$ and $g:\{a, b, e\} \rightarrow\{a, c\}$ be given by $f=\{(a, b),(c, e),(d, a)\}$ and $g=\{(a, c),(b, c),(e, a)\}$ write down
(i) gof
(ii) fog
Sol :
Question 5
Let $A=\{a, b, c, d\}, B=\{p, q, r, s\}, C=\{x, y, z\}$
Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be defined as
$f(a)=p, f(b)=q, f(c)=r, f(d)=s$
and $g(p)=x, g(q)=x, g(r)=y, g(s)=z$
Find gof $: A \rightarrow C$
Sol :
Question 6
Let $f:\{2,3,4,5\} \rightarrow\{3,4,5,9\}$ and
$g:\{3,4,5,9\} \rightarrow\{7,11,15\}$ be functions defined as
$f(2)=3, f(3)=4, f(4)=f(5)=5$
and $g(3)=g(4)=7$ and $g(5)=g(9)=11$.
Find gof.
Sol :
Question 7
If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=\cos x$ and $g(x)=3 x^{2}$, find gof and fog and show that gof ≠ fog
Sol :
Question 8
If $f(x)=8 x^{3}$ and $g(x)=x^{1 / 3}$, find $g o f$ and $f o g$.
Question 9
Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=x^{2}+2 x-3, g(x)=3 x-4$, find $(g o f)(x)$ and $(f o g)(x)$.
Question 10
Let $R$ be the set of all real numbers. If the functions $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=x^{2}+x+1$ and $g(x)=2 x$ for all $x \in R$, find
(i) fog
(ii) gof
(iii) fof
(iv) gog.
Question 11
Consider $f: N \rightarrow N, g: N \rightarrow N$ and $h: N \rightarrow R$ defined as $f(x)=2 x, g(y)=3 y+4$ and $h(z)=\sin z, \forall x, y, z \in N$. Show that ho $(g o f)=(h o g)$ of.
Question 12
Let $f: R \rightarrow R$ be defined by $f(x)=2 x-3$ and $g: R \rightarrow R$ be defined by $g(x)=\frac{x+3}{2}$
Show that $f o g=I_{R}=g o f$.
Question 13
Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be two one-one functions. Show that $g o f$ is also a one-one function.
Question 14
If $f(x)=x+7$ and $g(x)=x-7, x \in R$, find $(f o g)(7)$.
Question 15
If the function $f: R \rightarrow R$ be given by $f(x)=x^{2}+2$ and $g: R \rightarrow R$ be given by $g(x)=\frac{x}{x-1}, x \neq 1$, find fog and gof and hence find $f o g(2)$ and $g o f(-3)$.
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