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Question
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
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Solution
f : R → R is defined by f(x) = 3 – 4x
Let x1, x2 ∈ R such that f(x1) = f(x2)
⇒ 3 – 4x1 = 3 – 4x2
⇒ –4x1 = –4x2
⇒ x1 = x2
∴ f is one-one.
f : R → R be given for every y ∈ R (co-domain of f), there exists an element x ∈ R (domain of f) such that
f(x) = y
⇒ y = 3 – 4x
For any real number y ∈ R, there exists `x = (3 - y)/4 ∈ R` such that:
∴ `f((3 - y)/4)`
= `3 - 4 ((3 - y)/4)`
= 3 – 3 + y
= y
∴ f is onto.
Thus, f is one-one and onto and hence bijective.
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