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Question
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
Solution
Given, `f(x) = (2x - 7)/4`
For one-one
Let x, x2 ∈ R
f(x1) = f(x2)
`(2x_1 - 7)/4 = (2x_2 - 7)/4`
`\implies` 2x1 – 7 = 2x2 – 7
`\implies` 2x1 = 2x2
`\implies` x1 = x2
So, f(x) is a one-to-one function.
For onto
Put `y = (2x - 7)/4`
`\implies` 4y = 2x – 7
`\implies` 4y + 7 = 2x
`\implies` `x = (4y + 7)/2` ∀ y ∈ R ∃ a unique x ∈ R
Therefore, f(x) is onto.
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