**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".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Time and work word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**