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Question
If f : R → R is defined by f(x) = 3x − 5, prove that f is a bijection and find its inverse
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Solution
Given f(x) = 3x – 5
Let y = 3x – 5
y + 5 = 3x
⇒ `(y + 5)/3` = x
Let g(y) = `(y + 5)/3`
gof(x) = g(f(x))
= g(3x – 5)
= `(3x - 5 + 5)/3`
= `(3x)/5`
= x
gof(x) = x
fog(y) = f(g(y))
= `f((y + 5)/3)`
= `3((y + 5)/3) - 5`
= y + 5 – 5
fog(y) = y
∴ gof = Ix and fog = IY
Hence f and g are bijections and inverses to each other.
Hence f is a bijection and f-1(y) = `(y + 5)/3`
Replacing y by x we get f-1(x) = `(x + 5)/3`
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