classify as injection, surjection, or bijection(give reason):
f: R -> R, f(x)=x^3 - x
Answers
Step-by-step explanation:
it is bijection of couse
Given : f(x)=x³ - x
To find : classify as injection, surjection, or bijection
Solution:
f(x)=x³ - x
Let say
f(x₁) = f(x₂)
=> x₁³ - x₁ = x₂³ - x₂
=> x₁³ - x₂³ - x₁ + x₂ = 0
using a³ - b³ = (a - b)(a² + b² + ab)
=> (x₁ - x₂)(x₁² + x₂² + x₁x₂) - (x₁ - x₂) = 0
=> (x₁ - x₂) ( x₁² + x₂² + x₁x₂ - 1 ) = 0
=> ( x₁² + x₂² + x₁x₂ - 1 ) = 0
x₁ = - 1 & x₂ = 1 or vice versa satisfies this
=> f(1) = f(-1) = 0
but 1 ≠ - 1
Hence function is not injective
as its not injective hence its not bijective
f(x)=x³ - x is Continuous & differentiable hence its surjective
f(x)=x³ - x is surjective
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