Question

If the mapping of f and g is given by f(x) = {(1, 2), (3, 5), (4, 1)} and
g(x) = {(2, 3), (5, 1),(1, 3)} then write down the pair of fog and gof.

Answer 1

Hint:Now we are given mapping of f and g for three elements. Now we know (x, g(x)) and (x, f(x)). Now fog is nothing but f(g(x)). Hence for x we can first find g(x) then for this value we can find the output of f. that is f(g(x)). Similarly, for gof we will first find f(x) for any x and then for the value of f(x) we can check the output of g. hence we can write down the pair of fog and gof.

Complete step by step answer:
Now few are given that f(x) = {(1, 2), (3, 5), (4, 1)} and g(x) = {(2, 3), (5, 1),(1, 3)}.
Let us understand this
Now first consider f(x) = {(1, 2), (3, 5), (4, 1)}
Here each element is written as (x, f(x)).
This means f(1) = 2, f(3) = 5 and f(4) = 1.
Now consider g(x) = {(2, 3), (5, 1),(1, 3)}
Similarly here each element is (x, g(x)).
Hence we have g(2) = 3, g(5) = 1 and g(1) = 3.
Now let us consider fog. fog is a composite function defined as f(g(x)).
For example if we have x = 2, g(2) = 4 and f(4) = 3. Then we can say fog(2) = 3.
Now let us calculate fog with given data
Now consider g(2) = 3, g(5) = 1 and g(1) = 3.
Now we know that f(3) = 5, f(1) = 2 and f(3) = 5.
Hence we get
Now for x = 2 we have fog(2) = f(g(2)) = f(3) = 5
Hence for x = 2, fog(2) = 5
Now for x = 5 we have fog(5) = f(g(5)) = f(1) = 2
Hence for x = 5, fog(5) = 2
Now for x = 1 we have fog(1) = f(g(1)) = f(3) = 5
Hence for x = 1, fog(1) = 5
Hence we get fog = {(2,5), (5,2), (1,5)}
Now let us calculate gof.
Now let us calculate gof with given data.
Now consider g(2) = 3, g(5) = 1 and g(1) = 3.
Now we know that f(3) = 5, f(1) = 2 and f(3) = 5.
Hence we get
Now for x = 3 we have gof(3) = g(f(3)) = g(5) = 1
Hence for x = 3, gof(3) = 1
Now for x = 1 we have gof(1) = g(f(1)) = g(2) = 3.
Hence for x = 1, gof(1) = 3
Now for x = 3 we have gof(3) = g(f(3)) = g(5) = 1
Hence for x = 3, gof(1) = 1
Hence we get gof = {(3,1), (1,3)}

Note:
 Note that the composite function fog is defined as f(g(x)). Hence first we calculate g(x) and substitute the value in f(x). and for gof we will calculate g(x) and substitute the value in f(x). Do not make a silly mistake of doing it vice versa. Also note that $fog\ne gof$ .