Find the value

Question:

Let $A=\{1,2,3,4\} .$ Let $f: A \rightarrow A$ and $g: A \rightarrow A$,

defined by $f=\{(1,4),(2,1),(3,3),(4,2)\}$ and $g=\{(1,3),(2,1),(3,2),(4,4)\}$

Find (i) g of (ii) f o g (iii) f o f.

 

Solution:

(i) $\mathrm{g}$ o $\mathrm{f}$

To find: $g$ o $f$

Formula used: $g$ o $f=g(f(x))$

Given: $f=\{(1,4),(2,1),(3,3),(4,2)\}$ and $g=\{(1,3),(2,1)$

$(3,2),(4,4)\}$

Solution: We have,

$g \circ f(1)=g(f(1))=g(4)=4$

$g \circ f(2)=g(f(2))=g(1)=3$

$g \circ f(3)=g(f(3))=g(3)=2$

$g \circ f(4)=g(f(4))=g(2)=1$

Ans) g of $f=\{(1,4),(2,3),(3,2),(4,1)\}$

(ii) $f \circ g$

To find: f o g

Formula used: f o g = f(g(x))

Given: f = {(1, 4), (2, 1), (3, 3), (4, 2)} and g = {(1, 3), (2, 1),

(3, 2), (4, 4)}

Solution: We have,

fog(1) = f(g(1)) = f(3) = 3

fog(2) = f(g(2)) = f(1) = 4

fog(3) = f(g(3)) = f(2) = 1

fog(4) = f(g(4)) = f(4) = 2

Ans) f o g = {(1, 3), (2, 4), (3, 1), (4, 2)}

(iii) f o f

To find: $f$ o $f$

Formula used: $f$ o $f=f(f(x))$

Given: $f=\{(1,4),(2,1),(3,3),(4,2)\}$

Solution: We have,

fof $(1)=f(f(1))=f(4)=2$

fof $(2)=f(f(2))=f(1)=4$

fof $(3)=f(f(3))=f(3)=3$

fof $(4)=f(f(4))=f(2)=1$

Ans) fo f $=\{(1,2),(2,4),(3,3),(4,1)\}$ 

 

 

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