# How to find domain and range of fog and gof

**How to Find f o g and g o f From the Given Relation**

Definition :

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

Note : :

It should be noted that g o f exits if the range of f is a subset of g. Similarly, f o g 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 g o f.

**Answer :**

Write the function f as a set of ordered pairs.

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

Write the function g as a set of ordered pairs.

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

g o f can be denoted as

(g o f) (2) = 7

(g o f) (3) = 7

(g o f) (4) = 11

(g o f) (5) = 11

Therefore, g o f = {(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)}. Find g o f.

**Answer :**

f = {(1, 2) (3, 5) (4, 1)}

g = {(1, 3) (2, 3) (5, 1)}

g o f can be denoted as

(g o f) (1) = 3

(g o f) (3) = 1

(g o f) (4) = 3

Therefore, g o f = {(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 g o f and f o g are defined. Also find f o g and g o f.

**Answer :**

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 f o g is defined, then range of g must be a subset of domain of f. {3, 9} is a subset of {3, 9, 12}.

So, f o g is defined.

If g o f is defined, then range of f must be a subset of domain of g = {1, 3, 4} is a subset of {1, 3, 4, 5}.

So, g o f is defined.

g o f :

(g o f) (3) = 3

(g o f) (9) = 3

(g o f) (12) = 9

Therefore, g o f = {(3, 3), (9, 3), (12, 9)}.

f o g :

(f o g) (1) = 1

(f o g) (3) = 1

(f o g) (4) = 3

(f o g) (5) = 3

Therefore, f o g = {(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 g o f is defined while f o g is not defined. Also, find g o f.

**Answer :**

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

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

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

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

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

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

g o f :

(g o f) (1) = -2

(g o f) (4) = -4

(g o f) (9) = -6

(g o f) (16) = 8

Therefore, f o g = {(1, -2), (4, -4), (9, -6), (16, 8)}.

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## Question:

(a) Write formulas for {eq}displaystylefog text{ and } gof{/eq} and find the (b) domain and (c) range of each.

{eq}displaystyle1. f(x) = sqrt{(x + 1)}, g(x) = frac{1}{x} \2. f(x) = x^2, g(x) = 1 – sqrt{(x)}{/eq}

## Domain and Range of a Function:

A function is a relation between two sets defined by a definite rule. The set from which the values of the independent variables are taken, is called the domain of the function and the resultant set which is formed of the images of the domain variables is called the range set.

## Answer and Explanation:

1

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**1.)**

**Given:**

- The given functions are {eq}fleft( x right) = sqrt {left( {x + 1} right)}{/eq} and {eq}gleft( x right) =…

See full answer below.

Method and examples

**Composite functions and Evaluating functions : f(x), g(x), fog(x), gof(x) calculator**

Method 1. Find Range of `f:A->B`f(x) =A =

A

=,x `in` , x is

- `f(x)=5x+2` and `A={1<=x<5; x in N; x is odd}`
- `f(x)=3x-2` and `A={x in N}`
- `f(x)=|2x+1|` and `A={x in Z}`
- `f(x)=x^3` and `A={-2<=x<=2}`
- `f(x)=sqrt(x)` and `A={1,4,16,36}`
- `f(x)=x^2` and `A={[1,5]}`
- `f(x)=(-2)^x` and `A={x in N}`
- `f(x)=sin(x)` and `A={0,30,90}`
- `f(x)=(x^2+1)/(x+1)` and `A={-2,0,2}`
- `f(x)=x^2-2x+3` and `A={-1,3}`

2. Composite functions and Evaluating functions : f(2), g(3), fog(x), gof(x), fof(x), (f+g)(x)==Find()Also Find

- `f(x)=x-1, g(x)=3x+5`. Find fog(x).
- `f(x)=x+3, g(x)=x^2`. Find gof(x).
- `f(x)=3x+1, g(x)=-x^2+5`. Find gof(x).
- `f(x)=4x+1, g(x)=2x-3`. Find gof(2).
- `fog(x)=(x+2)/(3x), f(x)=x-2`. Find g(2).
- `gof(x)=(x+2)/(3x), g(x)=x-2`. Find f(x).
- `fog(x)=2x^2-1, f(x)=2+x^2`. Find g(x).
- `gof(x)=1/x^2, f(x)=2+x^2`. Find g(x).

3. Find valuef(x) =g(x) =h(x) =Find

- `f(x)=x(x+1)(2x+1)`. Find f(x)-f(x-1).
- `f(x)=1/(x+1)`. Find f(-x)-f(x).
- `f(x)=1/x`. Find f(x+1)-f(x-1).
- `f(x)=x^2-x`. Find f(x+1)-f(x).
- `f(x)=2x-3`. Find f(0)+f(1)+f(2).
- `f(x)=x^2-2^x`. Find f(2)-f(0).
- `f(x)=(x^2+1)/(x^3-x+1)`. Find f(1)-f(0).
- `f(x)=2x-6`. Find f(a-3)+f(a).
- `f(x)=(x-2)^2`. Find f(x+2).

4. Verifying if two functions are inverses of each otherf(x) =g(x) =

- `f(x)=x+3`. `g(x)=x-3`.
- `f(x)=4x-3`. `g(x)=(x+3)/4`.
- `f(x)=x/(x-1)`. `g(x)=(2x)/(2x-1)`.

Domain, Range, Inverse, Vertex, Symmetry, Directrix, Intercept, Parity, Asymptotes of a function calculator

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