Question

Check the injectivity and surjectivity of the function f :R → Rdefined by f(x) = 3 – 4x.



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Answer 1

SOLUTION

TO CHECK

The injectivity and surjectivity of the function f : R → R defined by f(x) = 3 – 4x

EVALUATION

CHECKING FOR INJECTIVE

Here the given function is

f : R → R defined by f(x) = 3 – 4x

Let a , b ∈ R such that f(a) = f(b)

Now f(a) = f(b)

⇒3 - 4a = 3 - 4b

⇒ - 4a = - 4b

⇒a = b

f(a) = f(b) gives a = b

So f is injective

CHECKING FOR SURJECTIVE

Let us take an arbitrary element y in the co-domain set R and let us examine if there is a pre-image x of the element y under f

Then f(x) = y

$\displaystyle \sf{ \implies \: 3 - 4x = y}$

$\displaystyle \sf{ \implies x = \frac{3 - y}{4} }$

Since y is arbitrary

So each element in the Co-domain set R has a pre-image under

So f is surjective

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