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Question
Following the case, state whether the function is one-one, onto, or bijective. Justify your answer.
f : R → R defined by f(x) = 3 − 4x
Solution
f: R → R is defined as f(x) = 3 − 4x.
Let `x_1 , x_2 in R " such that " f(x_1) = f(x_2)`
`=> 3 - 4x_1 = 3 - 4x_2`
`=> -4x_1 = -4x_2`
`=> x_1 = x_2`
∴ 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) in R, there `(3-y)/4` in R such that
`f((3-y)/4)`
= `3 -4 ((3-y)/4)`
= y
∴f is onto.
Hence, f is bijective.
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