Question

if f and g are onto then the function (gof)is?​

Answer 1

Given: f and g functions are onto functions.

To find: function (fog)

Step-by-step explanation:

Step 1 of 2

For a onto function fog, the range of fog will be equal to the codomain of f of g(x).

$Range(fog(x))=Codomainf(g(x))$

It means that,

$Range of fog(x)= Codomain of f(x)$

Step 2 of 2

f(x) is a superset of fog(x). Also the range of f(x) will be a subset of codomain of f(x).

$Range of f(x)$⊇ $Range of fog(x)$

$Range of f(x)$⊆ $Codomain of f(x)$

Therefore, $Range of f(x)=Codomain of f(x)$.

Hence, f(x) is onto.

Answer 2

If f and g are onto then the function (gof) is onto

Given :

The functions f and g are onto

To find :

The function (gof) is

Solution :

Step 1 of 2 :

Write down the given functions

Let f : A → B and g : B → C are onto

Step 2 of 2 :

Check the function (gof) is onto or not

Let z ∈ C

Since g is onto

There exists y ∈ B such that g(y) = z

Since f is onto

There exists x ∈ A such that f(x) = y

Thus we get

(gof)(x)

= g(f(x))

= g(y)

= z

Hence (gof) is onto

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