SOLUTION: Determine analytically, if the function 𝑓, whose correspondence rule is 𝑓(𝑥) = 𝑥^3 is injective

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Question 1194819: Determine analytically, if the function 𝑓, whose correspondence rule is 𝑓(𝑥) = 𝑥^3
is
injective
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given: f%28x%29+=+x%5E3

To show that a function is injective, we assume that there are elements a and b of a domain with f%28a%29+=+f%28b%29 and then show that a+=+b.

f%28a%29=f%28b%29
f%28a%29=a%5E3+
f%28b%29=b%5E3+
a%5E3=b%5E3 take the cubic root of each side
a=b
Therefore, f is injective.

Answer by ikleyn(52799) About Me  (Show Source):
You can put this solution on YOUR website!
.

Another way to build a proof is to notice that function   f(x) = x%5E3   is monotonic.

Monotonic function is always injective.