**How to Find fog and gof from the Given Relation ?**

Here we are going to see, how to find fog and gof from the given relation.

**Definition :**

Let f : A -> B and g : B -> C be two functions. Then a function gof : A -> C defined by (gof)(x) = g (f(x)), for all x ∈ A is called the composition of f and g.

**Point to remember :**

It should be noted that gof exits if the range of f is a subset of g. Similarly, fog exists if range of g is a subset of domain f.

**Question 1 :**

Let f : [2, 3, 4, 5] -> [3, 4, 5, 9] and g : [3, 4, 5, 9] -> [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.

**Solution :**

Writing the function "g" as a set of ordered pairs, we get

g = {(3, 7) (4, 7) (5, 11) (9, 11)}

Writing the function "f" as a set of ordered pairs, we get

f = {(2, 3) (3, 4) (4, 5) (5, 5)}

If gof exists, then range of "f" must be a subset of domain of "g".

Range of f = {3, 4, 5} and Domain of g = {3, 4, 5, 9}

From the picture given above, we know that

- 2 is associated with 3 and that is associated with 7. So, the result will be (2, 7).
- 3 is associated with 4 and that is associated with 7. So, the result will be (3, 7).
- 4 is associated with 5 and that is associated with 11. So, the result will be (4, 11).
- 5 is associated with 5 and that is associated with 11. So, the result will be (5, 11).

Hence the value of gof is = {(2, 7) (3, 7) (4, 11) (5, 11)}

**Question 2 :**

Let f : {1, 3, 4} -> [1, 2, 5] and g : [1, 2, 5] -> [1, 3] be given by f = {(1, 2) (3, 5) (4, 1)} and g = {(1, 3) (2, 3) (5, 1)}. Write down gof.

**Solution :**

**If gof exists, then range of "f" must be subset of domain of g.**

Range of f = {1, 2, 5}

Domain of g = {1, 2, 5}

From the picture given above, we know that

- 1 is associated with 2 and that is associated with 3. So, the result will be (1, 3).
- 3 is associated with 5 and that is associated with 1. So, the result will be (3, 1).
- 4 is associated with 1 and that is associated with 3. So, the result will be (4, 3).

Hence the value of gof is = {(1, 3) (3, 1) (4, 3)}

**Question 3 :**

Let f = {(3, 1) (9, 3) (12, 4)} and g = {(1, 3) (3, 3) (4, 9) (5, 9)}. show that gof and fog are defined. Also find fog and gof.

**Solution :**

f = {(3, 1) (9, 3) (12, 4)}

Domain of f = {3, 9, 12} and Range of f = {1, 3, 4}

g = {(1, 3) (3, 3) (4, 9) (5, 9)}

Domain of g = {1, 3, 4, 5} and Range of g = {3, 9}

If fog is defined, then range of "g" must be a subset of domain of "f". So, fog is defined.

If gof is defined, then range of "f" must be a subset of domain of "g".So, gof is defined.

**Finding gof :**

From the picture given above, we know that

- 3 is associated with 1 and that is associated with 3. So, the result will be (3, 3).
- 9 is associated with 3 and that is associated with 3. So, the result will be (9, 3).
- 12 is associated with 4 and that is associated with 9. So, the result will be (12, 9).

Hence the value of gof is {(3, 3) (9, 3) (12, 9)}

**Finding gof :**

From the picture given above, we know that

- 1 is associated with 3 and that is associated with 1. So, the result will be (1, 1).
- 3 is associated with 3 and that is associated with 1. So, the result will be (3, 1).
- 4 is associated with 9 and that is associated with 3. So, the result will be (4, 3).
- 5 is associated with 9 and that is associated with 3. So, the result will be (5, 3).

Hence the value of fog is {(1, 1) (3, 1) (4, 3) (5, 3)}.

**Question 4 :**

Let f = {1, -1) (4, -2) (9, -3) (16, 4)} and g = {(-1, -2) (-2, -4) (-3, -6) (4, 8)}. show that gof is defined while fog is not defined. Also, find gof.

**Solution :**

**f = {1, -1) (4, -2) (9, -3) (16, 4)}**

**Domain of f = {1, 4, 9, 16}, range of f = {-1, -2, -3, 4}**

** g = {(-1, -2) (-2, -4) (-3, -6) (4, 8)}.**

**Domain of g = {-1, -2, -3, 4}, range of g = {-2, -4, -6, 8}**

Since range of "f" is a subset of domain of g, gof is defined.

gof = {(1, -2) (4, -4) (9, -6) (16, 8)}

Since range of "g" is not a subset of domain of f, fog is not defined.

After having gone through the stuff given above, we hope that the students would have understood "How to Find fog and gof from the Given Relation".

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