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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